SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

SATURATED FUSION SYSTEMS OF
ESSENTIAL RANK 1
School of Mathematics
by
ELLEN HENKE
A thesis submitted to
The University of Birmingham
for the degree of
MASTER OF PHILOSOPHY (SC, QUAL)
The University of Birmingham
August 2009

Abstract
In this thesis we primarily consider saturated fusion systems which are of essential
rank 1, i.e. which contain only one conjugacy class of essential subgroups. We prove
that they all can be realized by an amalgam of two finite groups. This leads us to the
application of amalgam related methods like pushing up techniques and results about
FF-modules.
As a first key result, in fusion systems of essential rank 1 satisfying certain minor
extra conditions, we identify SL2(q) acting on a natural module inside the normalizer
of an essential subgroup. Note that there still is no bound on the number of non-trivial
chief factors inside an essential subgroup. This suggests that we need some stronger
assumptions and motivates us to introduce minimal fusion systems. Here our definition
is natural in the sense that every p-reduced fusion system contains a non-trivial section
which is minimal.
Expecting that the essential rank of a minimal fusion system is restricted, we again
focus on the case of essential rank 1. We then obtain detailed information about the
normalizer of an essential subgroup. If p = 2, this leads to a classification of the whole
fusion system, showing that the arising examples are all non-exotic.

Acknowledgements
First of all I would like to thank my supervisor Professor Sergey Shpectorov who gave
me the basic idea for this project and at the same time encouraged me to develop my
own ideas. I am very grateful to him for many fruitful discussions as well as for his
careful reading of parts of an earlier draft of this thesis.
Also, I am very thankful to Professor Bernd Stellmacher for his encouragement
and mathematical support. It was him who suggested to me the application of results
from [8]. Moreover, several arguments which appear in Chapters 8, 9 and 10 go back
to ideas from him. Apart from that, I thank him for his thorough reading of, and sug-
gestions regarding Chapters 4, 8, 9 and 10 of this thesis.
It is also a pleasure to thank Professor Chris Parker for many useful discussions.
Furthermore, I am very grateful to Miss Sarah Astill, who spend an awful lot of time
on correcting my language mistakes. She also suggested some mathematical improve-
ments to me regarding Chapters 2 and 3.
Finally, I thankful acknowledge financial support from the EPSRC.
2

Contents
1 Introduction 6
2 Preliminaries on Groups 15
2.1 Notation and basic results . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Coprime action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Dihedral and semidihedral 2-groups . . . . . . . . . . . . . . . . . . 21
2.5 Linear groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 2-Dimensional linear groups . . . . . . . . . . . . . . . . . . . . . . 29
2.7 Strongly p-embedded subgroups . . . . . . . . . . . . . . . . . . . . 32
2.8 Minimal parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . 36
2.9 Uniqueness of amalgams . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Groups Acting on Modules 43
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Offender . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Natural Sn-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Natural SL2(q)-modules . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 Recognizing SL2(q) . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Pushing Up 62
4.1 Applying a classical result . . . . . . . . . . . . . . . . . . . . . . . 62
3

4.2 The Baumann subgroup . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 Preliminaries on Fusion Systems 67
5.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Saturated fusion systems . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3 Normalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 Factor systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.5 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.6 Alperin–Goldschmidt’s Fusion Theorem . . . . . . . . . . . . . . . . 77
5.7 Minimal fusion systems . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 Amalgams Realizing Fusion Systems 87
6.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 Constructing Amalgams . . . . . . . . . . . . . . . . . . . . . . . . 88
7 Examples for p = 2 91
7.1 G ∼ = P GL2(q) for some odd q . . . . . . . . . . . . . . . . . . . . . 91
7.2 S semidihedral, G ∼ = P SL2(q 2 ) : 2 for some odd q . . . . . . . . . . 92
7.3 G ∼ = Aut(P SL2(9)) ∼ = Aut(A6) . . . . . . . . . . . . . . . . . . . . 92
7.4 P SL3(4) with the contragredient automorphism . . . . . . . . . . . . 94
8 Recognizing SL2(q) in Fusion Systems 98
8.1 Notation and main results of this chapter . . . . . . . . . . . . . . . . 98
8.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.3 The proof of Theorem 8.1.1 . . . . . . . . . . . . . . . . . . . . . . . 106
9 Pushing Up in Fusion Systems 107
9.1 Notation, basic properties and main results of this chapter . . . . . . . 107
9.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4

9.3 The proof of Lemma 9.1.3 . . . . . . . . . . . . . . . . . . . . . . . 116
9.4 The proof of Corollary 9.1.4 . . . . . . . . . . . . . . . . . . . . . . 124
9.5 The proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . 124
10 A Classification for p = 2 127
10.1 Bounding the 2-structure . . . . . . . . . . . . . . . . . . . . . . . . 127
10.2 Identifying F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5

Chapter 1
Introduction
Let G be a finite group, p a prime dividing the order of G and S a Sylow p-subgroup
of G. An old and important topic in finite group theory is fusion of subsets of S. Here
two subsets of S are said to be fused if they are conjugate in G. Regarding this, Alperin
published in [1] a fundamental result according to which fusion in S is controlled in
the normalizers of certain non-trivial subgroups of S. Thus the p-local subgroups of G,
i.e. the normalizers of non-trivial p-subgroups of G, basically provide all information
about fusion. On the other hand, most of the structure of p-local subgroups of G is
determined by the fusion in G.
Alperin’s result became known as Alperin’s Fusion Theorem. Goldschmidt pub-
lished in [13] a significant improvement of Alperin’s result further restricting the set
of subgroups, whose normalizers control fusion. We refer to Goldschmidt’s result
and similar versions of Alperin’s Fusion Theorem as Alperin–Goldschmidt’s Fusion
Theorem. In [20] Stellmacher used a graph theoretic approach to reprove Alperin–
Goldschmidt’s Fusion Theorem. He strengthened it by showing that the set of sub-
groups of S described by him is minimal possible.
Note that conjugacy of elements of a group plays a particularly important role in
character theory since conjugate elements have the same character value. Puig intro-
duced categories that he called full Frobenius systems when he was actually interested
6

in studying p-blocks of finite groups. Later these categories became known as satu-
rated fusion systems. The now standard notation and terminology was introduced by
Broto, Levi and Oliver in [7]. We define these in Chapter 5.
A fusion system F on an arbitrary finite p-group S is a category whose objects are
all subgroups of S and whose morphisms are group monomorphisms. Among these
morphisms are all conjugations by elements of S. For example if S is a subgroup of a
(not necessarily finite) group H then we can form a fusion system FS(H) on S where
the morphisms are just the maps induced by conjugation with elements of H.
In saturated fusion systems some extra properties hold which mimic the features
of Sylow subgroups in finite groups. So the fusion systems FS(G) where G is a finite
group and S a Sylow p-subgroup of G provide the main examples of saturated fusion
systems. However, there are saturated fusion systems which cannot be obtained in this
way. These fusion systems are called exotic.
From now on, for the remainder of the introduction, fusion systems are assumed to
be saturated and F is supposed to be a fusion systems on a finite p-group S.
Many exotic examples have been discovered so far. At first the search was some
what random. Later more systematic approaches were used which then led to the dis-
covery of new exotic examples. For instance, Ruiz and Viruel classified in [19] all
fusion systems on extraspecial groups of order p 3 and exponent p, and found that some
of these, for p = 3, 5, 7 and 13, were exotic.
Anyway, this choice of a p-group seems to us rather random as in fact any other
choice might be. Therefore we believe it is now time to investigate classes of fusion
systems which can be described in the fusion theoretic language rather than just in
terms of the underlying groups. Naturally we also expect this to lead to the discovery
of new exotic examples, as well as to the development of more structure theory for
7

fusion systems.
A significant step in this direction has already been made by Aschbacher in [3],
where he obtained classification results for fusion systems of characteristic 2-type, i.e.
fusion systems over 2-groups in which the normalizers of fully normalized subgroups
are constrained. During his talk at the Isle of Skye conference in June 2009, Asch-
bacher also announced some results about fusion systems of component type.
Aschbacher’s aim is to improve the classification of finite simple groups by treating
problems in the larger category of fusion systems. As a consequence of this, many of
his concepts are motivated by analogous concepts in finite group theory which were
crucial for the classification of finite simple groups. Also, this is why he restricts him-
self to fusion systems over 2-groups. Moreover he needs to assume that the groups of
F-automorphisms are K-groups. Here a finite group is a K-group if each of its simple
sections is a “known” simple group appearing in the statement of the theorem classi-
fying the finite simple groups.
We want to stress at this point that for the problems considered in this thesis we
do not need any kind of K-group hypothesis. Furthermore we obtained some fairly
strong results for odd primes as well. Also, as far as we know, our main concepts do
not have any analogs of considerable importance in finite group theory. Significant to
our approach is what we call the essential rank of a fusion system and a minimality
condition.
We first come to the definition of essential rank. Broto, Levi and Oliver published
in [7] a theorem about fusion systems which is similar to Alperin’s Fusion Theorem.
Probably, Puig was already aware of an even stronger result analogous to Alperin–
Goldschmidt’s Fusion Theorem. The version of that, which is important for us, states
that a set of subgroups of S is a conjugation family for F, if it contains S and inter-
8

sects non-trivially with every F-conjugacy class of essential subgroups. Here we call
a family of subgroups of S a conjugation family if fusion is controlled in the norma-
lizers of its members. A subgroup Q of S is said to be essential if Q is centric and
MorF(Q, Q)/Inn(Q) has a strongly p-embedded subgroup.
In fact, there is even a result on fusion systems which is roughly equivalent to
Stellmacher’s version of Alperin–Goldschmidt’s Fusion Theorem. A set of subgroups
of S is a conjugation family if and only if it contains S and intersects non-trivially
with every F-conjugacy class of essential subgroups. One direction of this result was
proved by Fyn-Sydney in [11], where she also characterized minimal conjugation fa-
milies. We learned from Shpectorov that the other direction holds as well. Since, as far
as we know, this is not published anywhere, we reprove this result formally in Section
5.6. The statement of Alperin–Goldschmidt’s Fusion Theorem as above shifts focus
from the particular essential subgroups to the F-conjugacy classes of essential sub-
groups. This motivates the following definition, which again was suggested to us by
Shpectorov.
Definition 1. The essential rank of F is the number of F-conjugacy classes of essential
subgroups in F.
Our general idea is to consider fusion systems of low essential rank. If F has
essential rank 0 then S is normal and centric in F. If a fusion system F has a normal
centric subgroup, then we call F constrained and it follows from a result of Broto,
Castellana, Grodal, Levi and Oliver in [6] that there exists a model for F. Here a
model for F is a finite group G of characteristic p containing S as a Sylow p-subgroup
and realizing F. Thus, in particular, there exists a model for F if F has essential rank
0. Moreover S will be normal in such a model. On the other hand, each finite group
with a normal Sylow p-subgroup leads to a saturated fusion system of essential rank 0.
Therefore we have completely classified them. This is why we consider essential rank
9

1 to be the first interesting case. We now start to state our results.
Unless otherwise indicated assume from now on that F has essential rank 1 and
Q is essential in F. Note that, as a consequence of Alperin–Goldschmidt’s Fusion
Theorem, Q is then fully normalized in F. In Section 6.2 we prove the following:
Proposition 1. There exists an amalgam (G1, G2, G12, φ1, φ2) with the following prop-
erties:
(a) G1 is a model for NF(S) and G2 is a model for NF(Q).
(b) NS(Q) ≤ G12 and both φ1 and φ2 are the identity on NS(Q). Moreover G12φ1 =
NG1(Q) and G12φ2 = NG2(NS(Q)).
(c) G12φ2/Q is a strongly p-embedded subgroup of G2/Q.
(d) F = FS(H) for H = G1 ∗G12 G2.
Here an amalgam A is a tuple (G1, G2, G12, φ1, φ2) where G1, G2 and G12 are
groups and φi : G12 → Gi is a monomorphism for i = 1, 2. We write G1 ∗G12 G2 for
the free product of G1 and G2 with G12 amalgamated. Note that we suppress mention
of the monomorphisms φ1 and φ2.
From now on assume that A = (G1, G2, G12, φ1, φ2) is the amalgam constructed
in Proposition 1. Moreover set
H = G1 ∗G12 G2,
T = NS(Q),
M = O p′
(G2).
Here Op′ (G2) denotes the smallest normal subgroup of G2 with a factor group of order
coprime to p. Under some minor assumptions we obtain fairly strong information
about the structure of G2. In order to state it we need one more definition.
10

Definition 2. Let V be a finite-dimensional GF (p)G-module and G ∼ = SL2(q) for
some power q of p. Then V is called a natural SL2(q)-module for G if V is irreducible,
F := EndG(V ) ∼ = GF (q) and V is a 2-dimensional F G-module.
Here by EndG(V ) we denote the set of all GF (p)-linear transformations of V
which commute with the action of G on V . Note that by Schur’s Lemma and Wedder-
burn’s Theorem, EndG(V ) is a finite field whenever V is irreducible. We prove the
following theorem.
Theorem 1. Assume Ω(Z(S)) and J(S) are not normal in F. Let V be a normal
subgroup of M such that Ω(Z(S)) ≤ V ≤ Ω(Z(Q)). Then the following hold:
(a) NS(J(Q)) = T , J(T ) �≤ Q and CS(V ) = Q.
(b) CG2(V )/Q is a p ′ -group.
(c) For some power q of p, M/CM(V ) ∼ = SL2(q) and V/CV (M) is a natural
SL2(q)-module for M/CM(V ).
In fact we prove a slightly more general result (see Theorem 8.1.1), which also
applies in certain cases of larger essential rank. Furthermore in Section 8.2, we obtain
a number of technical lemmas which might be of interest in some situations.
In the proof of Theorem 1 we apply results from [8] (see Section 3.5 and Chapter
8 of this thesis). This makes it possible to avoid the use of a K-group assumption.
In order to state the final two results we need to introduce the concept of minimality.
Assume for the moment that F has arbitrary essential rank.
Definition 3. We call a fusion system F on S minimal if Op(F) = 1 and for every
non-trivial fully normalized subgroup U of S either NF(U) has essential rank 0 or
Op(NF(U)/Op(NF(U))) �= 1.
11

We call F strongly minimal if Op(F) = 1 and for every non-trivial fully normalized
subgroup U of S, NF(U)/Op(NF(U)) has essential rank 0.
Note here that a fusion system has essential rank 0 if and only if its underlying
p-group is normal. Hence every strongly minimal fusion system is minimal.
The idea to consider strongly minimal fusion systems was suggested to us by Sh-
pectorov who called these systems minimal. However, what we would like to achieve
is that a fusion system F with Op(F) = 1 has a non-trivial section which is minimal.
This is definitely true with the definition of minimality as above. Unfortunately we are
not able to prove an analogous result for strong minimality.
Minimal fusion systems do not necessarily need to have essential rank 1 as we
know examples of strongly minimal fusion systems of rank 2. Anyway, we hope that
the essential rank of minimal systems, or at least of strongly minimal systems, is re-
stricted.
In this thesis we focus on minimal fusion systems of essential rank 1. In order
to state the final two results we again assume that F has essential rank 1 and use the
notation introduced above. We prove the following theorem.
Theorem 2. Let F be minimal and let q be as in Theorem 1. Then NS(X) = T for
every normal p-subgroup X of M. Furthermore, one of the following holds:
(I) Q is elementary abelian, M/Q ∼ = SL2(q) and Q/CQ(M) is a natural SL2(q)-
module for M/Q, or
(II) p = 3, |Q| ≤ q 6 and S = T . Moreover there is a subgroup H of M such that
T ≤ H, M = CM(Ω(Z(Q)))H and for V := [Q, O p (H)], Z(V ) ≤ Z(Q),
V/Z(V ) and Z(V )/Φ(V ) are natural SL2(q)-modules for H/CH(V (Q)), and
Φ(V ) = CV (M) has order q.
Moreover, if (I) holds and p = 2 or F is strongly minimal, then |Q| ≤ q 3 .
12

In order to show Theorem 2 we apply Theorem 1 and a pushing-up result, which
was shown by Baumann in [5], Niles in [17] and later improved by Stellmacher in [21].
Apart from that, our proof is self contained. If S is a 2-group then Theorem 2 leads to
the following complete classification.
Theorem 3. Let F be minimal and p = 2. Let q be as in Theorem 1. Then q ≤ 4.
Moreover F ∼ = FS(G) for some finite group G such that S ∈ Syl2(G) and one of the
following holds:
(a) q = 2, S is dihedral of order at least 16 and G = P GL2(r) for some odd prime
power r.
(b) q = 2, S is semidihedral of order at least 16 and for some odd prime power
r, G is the unique extension of L2(r 2 ) of degree 2 with semidihedral Sylow 2-
subgroups.
(c) q = 2, |S| = 2 5 and G = Aut(A6).
(d) q = 4, |S| = 2 7 and G is the semidirect product of L3(4) with the contragredient
automorphism.
In order to deduce Theorem 3 from Theorem 2 we classify the amalgam A realizing
F. The arguments used there are probably not entirely new. However, in order to keep
things reasonably self-contained we prove everything directly as it is often difficult or
impossible to find suitable references.
The groups listed in Theorem 3 all occur in Aschbacher’s theorem about fusion
systems of characteristic 2-type. We are not sure at the moment how our assumptions
are related to Aschbacher’s hypothesis of local characteristic 2. There might be a way
to obtain our result as a corollary of Aschbacher’s result, although our approach still
would have the advantage of avoiding any kind of K-group hypothesis.
13

Even if there is no direct relationship between our result and Aschbacher’s theorem,
there is a connection regarding the methods in the proofs. This is first of all, because
we mostly work within constrained p-local subsystems. And secondly, the strategies
and results we use all come from the classification of finite simple groups of local
characteristic 2.
14

Chapter 2
Preliminaries on Groups
Throughout this thesis we write function on the right side. In this chapter, if φ is a
homomorphism from a group G, we usually write g φ rather than gφ for the image of g
under φ. Similarly, we write G φ for the image of G under φ.
The aim of this chapter is to give an overview on some basic group theoretical
background and to fix some notation as necessary. In that we follow [15]. We also
prove some more specialized results which we will need later on. In particular, in
Section 2.9 we show the uniqueness of certain amalgams.
In the remainder of this chapter G will always be a finite group, p a prime and T a
Sylow p-subgroups of G. Moreover, q is always assumed to be a power of p.
2.1 Notation and basic results
We will write o(g) for the order of an element g ∈ G, and Sylp(G) for the set of Sylow
p-subgroups of G. The group G is called p-closed if T is normal in G. We will make
use of the following characteristic subgroups of G:
• Op(G) is the largest normal p-subgroup of G.
• O p (G) is the smallest normal subgroup of G whose factor group is a p-group.
Equivalently, O p (G) is the group generated by all p ′ -elements of G.
15

• Op′ (G) is the smallest normal subgroup of G of index prime to p. Note that
Op′ (G) = 〈Sylp(G)〉.
• Op (Op′ (G)) is the smallest normal subgroup of G with a p-closed factor group.
• If G is a p-group then Ω(G) is the subgroup generated by all elements of G
of order p. If G is also abelian then Ω(G) is the largest elementary abelian p-
subgroup of G.
• We write Φ(G) for the Frattini subgroup of G, which is the intersection of all
maximal subgroups of G. If G is a p-group then Φ(G) is the smallest normal
subgroup of G with an elementary abelian factor group.
Definition 2.1.1. G has characteristic p if CG(Op(G)) ≤ Op(G).
If N is a normal subgroup of G, then we will often make use of the so called “bar”-
notation. This means that, after setting G = G/N, we write U instead of UN/N for
every subgroup U of G, and g instead of gN for every g ∈ G.
We assume the reader to be familiar with Sylow’s Theorem. Moreover we will
frequently use that p-groups are nilpotent. In particular, if G is a p-group, then U <
NG(U) for every proper subgroup U of G and [N, G] < N for every normal subgroup
N of G.
If A, B and C are subgroups of G such that G = AB and A ≤ C, then C =
A(C ∩ B). We will refer to this property as Dedekind’s Law.
If H, A are subgroups of G then A is called a complement of H in G if G = HA
and H ∩ A = 1. If H is in addition normal in G, we call G the (internal) semidirect
product of A with H and write G = A ⋉ H.
On the other hand, if A is a group acting on a group H, then we can form the
(external) semidirect product of A and H denoted by either A ⋉ H or AH. The
16

groups A and H can be identified with subgroups of AH, and then AH is actually also
the internal semidirect product of A with H. Moreover, for every h ∈ H and a ∈ A,
the image h a of h under a via the action of A on H is just the same as a −1 ha, i.e. as
the conjugate of h by a inside the semidirect product AH. Note that AH is uniquely
determined up to isomorphism by the structure of A, H and the action of A on H.
More precisely, we have
Remark 2.1.2. Let H, ˜ H, A and à be finite groups such that A acts on H, and à acts
on ˜ H. Suppose there are group isomorphism φ : H → ˜ H and ψ : A →
à such
that (h a )φ = (hφ) (aψ) for all h ∈ H and a ∈ A. Then there is a group isomorphism
α : AH → Ã ˜ H such that α|H = φ and α|A = ψ.
As a consequence of this we get
Remark 2.1.3. Let H be a finite group, A ≤ Aut(H) and φ ∈ Aut(H). Then there is
a group isomorphism α : AH → (φ −1 Aφ)H such that α|H = φ and aα = φ −1 aφ for
all a ∈ A.
Assume from now on that G acts on a set Ω and α ∈ Ω. Then we write α G for the orbit
of α under G and |α G | for the length of this orbit. The stabilizer of α in G is denoted
by Gα. We will frequently use that |G : Gα| = |α G |. Moreover, we will apply the
following lemma:
Lemma 2.1.4 (Frattini Argument). Let N be a subgroup of G acting transitively on Ω.
Then G = GαN for every α ∈ Ω.
Using Sylow’s Theorem we get as an immediate consequence of this lemma that for
a normal subgroup N of G and P ∈ Sylp(N), G = NNG(P ). We will refer to this
property as Frattini Argument as well.
Definition 2.1.5. The action of G on Ω is said to be 2-transitive if |Ω| ≥ 2 and G acts
transitively on the set Ω (2) := {(α, β) ∈ Ω × Ω : α �= β}. Here the action of G on
Ω (2) is defined by (α, β) g = (α g , β g ) for all (α, β) ∈ Ω (2) and all g ∈ G.
17

2.2 Commutators
For elements x, y ∈ G we set
[x, y] := x −1 y −1 xy
and call [x, y] the commutator of x and y.
Remark 2.2.1. For every group homomorphism φ from G and all x, y ∈ G we have
[x, y] φ = [x φ , y φ ]. In particular [x, y] g = [x g , y g ] for all g ∈ G.
In order to make things more convenient we define
[x, y, z] := [[x, y], z] for all x, y, z ∈ G.
Moreover, for subsets X, Y, Z of G, we set
[X, Y ] = 〈[x, y] : x ∈ X, y ∈ Y 〉,
[X, Y, Z] = [[X, Y ], Z].
Note that [x, y] −1 = [y, x] for all x, y ∈ G. In particular, [X, Y ] = [Y, X] for all
subsets X and Y of G.
Many elementary properties inside groups can be expressed using commutators.
• For x, y ∈ G we have
• For subgroups X, Y of G
[x, y] = 1 ⇐⇒ xy = yx ⇐⇒ x y = x.
[X, Y ] ≤ Y ⇐⇒ [Y, X] ≤ Y ⇐⇒ Y is X-invariant
We will make use of the following commutator formula.
Lemma 2.2.2. [x, yz] = [x, z][x, y] z and [xz, y] = [x, y] z [z, y], for all x, y, z ∈ G.
18

Lemma 2.2.3. For subgroups X, Y of G, [X, Y ] is a normal subgroup of 〈X, Y 〉.
Proof. See for example [15, 1.5.5].
Lemma 2.2.4 (Three-Subgroup Lemma). Let X, Y, Z ≤ G and let N be a normal
subgroups of G. Assume [X, Y, Z] ≤ N and [Y, Z, X] ≤ N. Then [Z, X, Y ] ≤ N.
Proof. This follows from [15, 1.5.6].
If a group A acts on the group G, then we can consider at A and G as subgroups of
the semidirect product A ⋉ G. Thus, the definitions of commutators and commutator
subgroups can be inherited and the results hold accordingly.
In particular, the Three-Subgroup Lemma holds, i.e. if each of X, Y, Z is a sub-
group of either G or A then
2.3 Coprime action
[X, Y, Z] = [Y, Z, X] = 1 =⇒ [Z, X, Y ] = 1.
In this section A will always be a group which acts on G. We call the action of A on
G coprime if
(1) |A| and |G| are coprime, and
(2) A or G is soluble.
We remind the reader here that a finite group G is called soluble if [U, U] < U for
all subgroups U of G. Thus abelian groups and nilpotent groups provide examples of
soluble groups. In particular, p-groups are soluble.
By a Theorem of Feit-Thompson, every finite group of odd order is soluble. Hence
if |K| and |G| are coprime then either K or G is soluble. Therefore assumption (2)
is in fact redundant. However we prefer to put it this way since the Theorem of Feit-
Thompson is a deep result of finite group theory and in all the situations we are going
19

to apply the results about coprime action, assumption (2) is fulfilled for some other
reasons.
Lemma 2.3.1. Assume A acts coprimely on G. Then
(a) G = [G, A]CG(A),
(b) [G, A] = [G, A, A].
Proof. See for example [15, 8.2.7].
Lemma 2.3.2. Suppose G is abelian and A acts coprimely on G. Then G = [G, A] ×
CG(A).
Proof. See for example [15, 8.4.2]
Often we will need the following generalization of 2.3.1(b) which follows immediately
from 2.2.2 and 2.3.1(b).
Lemma 2.3.3. If A is generated by elements whose order is coprime to |G| then
[G, A] = [G, A, A].
In particular, if G is a p-group, then [G, O p (A)] = [G, O p (A), O p (A)]. We will use this
property frequently and usually without reference.
Lemma 2.3.4 (P × Q-Lemma of Thompson). Let A = P × Q be the direct product of
a p-group P with a p ′ -group Q. Assume that G is a p-group such that
Then [G, Q] = 1.
CG(P ) ≤ CG(Q).
Proof. A proof can be found for example in [15, 8.2.8].
We note a consequence of Thompson’s P × Q-Lemma which we will need in Section
6.2.
20

Lemma 2.3.5. Let G be a finite group and S a normal p-subgroup of G such that
CG(S) ≤ S. Assume there is Q ≤ S with CS(Q) ≤ Q. Then CG(Q) is a p-group. In
particular, if S ∈ Sylp(G) then CG(Q) ≤ Q.
Proof. Let R be a p-prime subgroup of CG(Q). Then A = R × Q acts on the p-group
S. Moreover, CS(Q) ≤ Q ≤ CS(R). Therefore, by Thompson’s P × Q-lemma,
[S, R] = 1. Hence, R ≤ CG(S) ≤ S and R = 1. This shows that CG(Q) is a p-group.
If S ∈ Sylp(G), we get now CG(Q) ≤ CS(Q) ≤ Q.
2.4 Dihedral and semidihedral 2-groups
We remind the reader that the group G is called dihedral if it is generated by two
involutions. Equivalently, there exist elements t, x ∈ G such that t is an involution,
G = 〈t〉 ⋉ 〈x〉 and x t = x −1 . We are particularly interested in the case where G is a
2-group. Note that we also regard fours groups as dihedral groups. A group which is
similar to the dihedral 2-group is the semidihedral 2-group, which is defined as follows.
Definition 2.4.1. G is called semidihedral, if there exist elements x, t ∈ G such that t
is an involution, o(x) = 2 n for some n ≥ 3, G = 〈t〉 ⋉ 〈x〉 and x t = x 2n−1 −1 .
While studying dihedral and semidihedral 2-groups, the automorphisms of cyclic 2-
groups play an important role. We summarize some of their properties in the following
lemma.
Lemma 2.4.2. Let X = 〈x〉 be a cyclic group of order 2 n , n ≥ 2.
(a) xσ2 �= x−1 for every σ ∈ Aut(X).
(b) Aut(X) has order 2 n−1 .
(c) If n ≥ 3 then the automorphisms α, β, γ of X defined by
x α = x −1 , x β = x 2n−1 −1 , x γ = x 2 n−1 +1
21

are the only involutions in Aut(X).
Proof. This follow from 2.2.5 and 2.2.6 in [15].
Lemma 2.4.3. Let G be a dihedral or semidihedral 2-group and G = 〈t〉 ⋉ 〈x〉 where
t, g ∈ G and t is an involution. Then the following properties hold:
(a) If G is dihedral then every element in G\〈x〉 is an involution.
(b) If G is semidihedral then every element in G\〈x〉 has order 2 or 4. More pre-
cisely, for each integer i, o(x i t) = 2 if i is even and o(x i t) = 4 if i is odd.
(c) 〈x〉 is characteristic in G.
(d) Aut(G) is a 2-group.
(e) If G is dihedral then G has exactly two dihedral subgroups of index 2.
(f) If G is semidihedral then G has exactly one dihedral subgroup of index 2.
(g) Aut(G) acts transitively on the set of subgroups of G isomorphic to D8.
(h) If D1 ≤ G is dihedral of order 8, then Aut(D1) ∼ = NG(D1)/CG(D1) or G = D1.
Proof. Parts (a) and (b) are shown by elementary calculations. Part (c) follows from
(a) and (b). Thus, Aut(G) acts on 〈x〉. If A ≤ Aut(G) has odd order then it follows
from 2.4.2(b) that A centralizes 〈x〉. Since |G/〈x〉| = 2, we have also [G, A] ≤ 〈x〉.
Hence, 2.3.1(b) implies A = 1 and (d) holds.
Let D be a dihedral subgroup of G of index 2. A calculation shows that, if G is
semidihedral of order 2 4 , then the only cyclic subgroup of order 4 which is normalized
by an involution is 〈x 2 〉. In any other case, 〈x 2 〉 is the only cyclic subgroup of index 4
in G. Therefore, 〈x 2 〉 ≤ D. If G is semidihedral then by (b), D = 〈x l t, x 2 〉 for some
even integer l. Then D = 〈x, t〉 and D is uniquely determined. This shows (f).
22

If G is dihedral, then D = 〈x 2 , x l t〉 for some arbitrary integer l. If l is even then
D = 〈x 2 , t〉, and if l is odd then D = 〈x 2 , xt〉. Since 〈x 2 , t〉 �= 〈x 2 , xt〉, this shows (e).
Let Y be a subgroup of 〈x〉 of order 4. Using (b), an elementary calculation shows
that every element of order 4, which is inverted by an involution from G, is contained in
Y . So every subgroup of G isomorphic to D8 is of the form 〈s〉⋉Y for some involution
s ∈ G\〈x〉. Note that Y is characteristic in G. Hence, for the proof of (f), it is suffi-
cient to show that Aut(G) acts transitively on the set of involutions in G\Z(G). Let
s ∈ G\Z(G) be an involution. Note that C〈x〉(s) = Z(G) and thus CG(s) = Z(G)〈s〉.
So |x G | = |G|/4. If G is semidihedral then it follows from (b) that G acts transitively
on the set of involutions in G\Z(G). This shows (f) in the semidihedral case. If G is
dihedral then G acts transitively on the set of involutions in D\Z(G) for any dihedral
subgroup D of G of index 2. Since G embeds into a dihedral group of order 2|G|, this
implies (f).
Now let D1 ≤ G be isomorphic to D8 and A = Aut(D1). Note that by (d), A is
a 2-group. By (e), D1 has exactly two subgroups Q, R that are fours groups. Then A
acts on {Q, R} and thus |A/NA(Q)| = 2. Moreover, Aut(Q) ∼ = Aut(R) ∼ = S3. Thus,
|NA(Q)/CA(Q)| ≤ 2 and |CA(R)| = |CA(R)/CA(R)∩CA(Q)| ≤ |NA(Q)/CA(Q)| ≤
2, since CA(R) ∩ CA(Q) = CA(D1) = 1. This shows |A| ≤ 8. Now note that
NG(D1)/CG(D1) embeds into A. Moreover, if D1 �= G, NG(D1) is dihedral or semidi-
hedral of order 16 and CG(D1) = Z(NG(D1)) has order 2. This implies (h).
Lemma 2.4.4. Let G be a 2-group and D0 ≤ G be dihedral of order 2 n , n ≥ 2.
Assume D1 := NG(D0) is dihedral of order 2 n+1 and normal in G. Then G is dihedral
or semidihedral and |G/D1| ≤ 2.
Proof. The group G acts on the set of dihedral subgroups of D1. Therefore, |G/D1| ≤
2. We may assume that |G/D1| = 2. Then there exists an element y ∈ G\D1. Note
that y 2 ∈ D1.
23

Let x, t ∈ D1 such that t is an involution, D1 = 〈t〉 ⋉ 〈x〉 and x t = x −1 . Moreover,
assume that D0 = 〈t〉 ⋉ 〈x 2 〉. Then by 2.4.3(c), y induces an automorphism on 〈x〉.
Thus, it follows from 2.4.2(a) that y 2 ∈ 〈x〉. If y 2 ∈ 〈x 2 〉 then
(yt) 2 = y 2 [y, t] ∈ 〈x 2 〉[y, t].
Note that 〈x 2 〉 is characteristic in 〈x〉 and therefore normalized by y. Since y �∈ D1 =
NG(D0), it follows that [t, y] �∈ 〈x 2 〉. Hence, if y 2 ∈ 〈x 2 〉, then (yt) 2 �∈ 〈x 2 〉. Thus, we
may assume that y 2 �∈ 〈x 2 〉. Then y has order 2 · o(x) = 2 n+1 and 〈y〉 has index 2 in
G. Hence, G = 〈t〉 ⋉ 〈y〉.
Assume y t = y 2n−1 +1 . Then (y 2 ) t = y 2 and therefore [x, t] = 1, a contradiction.
Thus, it follows from 2.4.2(c) that G is dihedral or semidihedral.
Lemma 2.4.5. Let G be a 2-group and Q ≤ G a fours group such that CG(Q) ≤ Q.
Then G is dihedral or semidihedral.
Proof. Set G0 = Q and Gi+1 = NG(Gi) for i ≥ 0. Since G is nilpotent, there exists
n ≥ 0 such that G = Gn.
Assume Q �= G. Then Q < G1. Since CG(Q) ≤ Q and Aut(Q) ∼ = S3, it follows
that |G1/Q| = 2. Moreover, T1 is not abelian and contains more than one involution.
Hence, G1 is dihedral of order 8.
Suppose, for some m ≥ 2, that Gm is semidihedral and Gm−1 is dihedral. Then by
2.4.3(f), Gm−1 is the only dihedral subgroup of Gm and therefore characteristic in Gm.
Thus, Gm+1 ≤ NG(Gm−1) = Gm and therefore G = Gm.
Now it follows from 2.4.4 and induction that for all 0 ≤ i < n, |Gi+1/Gi| = 2
and Gi+1 is dihedral or semidihedral. Thus, in particular, G = Gn is dihedral or
semidihedral.
Lemma 2.4.6. Let G ∼ = S4 × C2, T ∈ Syl2(G), Q = Op(G) and C = Z(G). Let
α, β ∈ Aut(T ) be involutions such that
24

(a) C α �= C �= C β ,
(b) Q α �= Q �= Q β .
Then there is h ∈ T and µ ∈ Aut(G) such that µ 2 = 1, h β = h −1 and α(β µ ) = ch,
where ch is the inner automorphism of T determined by h ∈ T .
Proof. Let γ ∈ {α, β} and set S = 〈γ〉 ⋉ T . We will identify γ and T with there
images in S. Note that Q ∩ Q a = Z(T ) for every a ∈ S\T . By (b), S = S/Z(T )
is non-abelian, since Q is not normal in S. In particular, there exists an element in S
which has order 4. Since T is elementary abelian, we can choose y ∈ S\T such that y
has order 4. Then y 4 ∈ Z(T ), and S = T 〈y〉 implies y 4 ∈ Z(S). If y 4 = 1 then y 2 ∈ T
is an involution. Observe that every involution in T is contained in Q or Q y . Hence,
we would have y 2 ∈ Q or y 2 ∈ Q y . However, then y 2 = (y 2 ) y ∈ Q ∩ Q y = Z(T ) and
y has order 2, a contradiction. Therefore, y 4 is an involution and y has order 8.
Since Z(T ) is normal in S, [C, 〈y〉] ≤ [Z(T ), 〈y〉] ≤ Z(T ). Moreover, it follows
from [C, 〈y 2 〉] = 1 and 2.2.2(a) that [C, 〈y〉] = 〈[c, y]〉 for 1 �= c ∈ C. Since Z(T ) is
elementary abelian, [c, y] ∈ Z(T ) has order 2, i.e. |[C, 〈y〉]| = 2. By 2.2.3, [C, 〈y〉]
is normalized and therefore centralized by y. Hence, [C, 〈y〉] ≤ Z(S) = 〈y 4 〉 and
〈y〉 is normalized by C. By (a), [C, y] �= 1. Now [y 2 , C] = 1 implies y c = y 5 .
Set N = 〈y〉C. Then 〈y〉 and 〈yc〉 are the only cyclic subgroups of N of order 8.
Moreover, |S : N| = 2 and so N is normal in S. Hence, q ∈ Q\Z(T ) acts on N and
either normalizes 〈y〉 or swaps 〈y〉 and 〈yc〉.
Assume first y q ∈ 〈yc〉 = 〈y 2 〉 ∪ 〈y 2 〉yc. Since y q �∈ T , y q = y i c for some
i ∈ {1, 3, 5, 7}. Then y 2 �= (y 2 ) q = (y q ) 2 = (y i c) 2 = y i (y i ) c = y i (y c ) i = y i y 5i = y 6i
implies i ∈ {1, 5}. Hence, [y, q] = y −1 y q ∈ Z(T ) ≤ Q and Q is normalized by y, a
contradiction to (b). Thus, 〈y〉 is normal in S and S is the semidirect product of 〈c, q〉
and 〈y〉. Moreover, y 2 is not centralized by q and so by 2.4.2(c), 〈c, q〉 acts faithfully
25

on 〈y〉. Since y c = y 5 , y j c is not an involution for any odd number j. Since γ is an
involution in S\T , it follows γ = y j t for some odd number j and some t ∈ 〈c, q〉 with
t �= c. Then 〈y, t〉 is dihedral or semidihedral and (y 2 ) γ = y −2 . Also, γ acts on Z(T ),
fixes y 4 , and does not fix C. Hence, c γ = y 4 c. We get then (y 2 c) γ = y −2 y 4 c = y 2 c.
Observe also that S/〈y 2 〉 is abelian and thus, [q, γ] ≤ 〈y 2 〉. Moreover, [q, γ] �≤ 〈y 4 〉,
since Q = Z(T )〈q〉 is not normalized by γ. Hence, q γ ∈ 〈y 4 〉yq. Also observe that the
only cyclic subgroups of T of order 4 are 〈y 2 〉 and 〈y 2 c〉.
Now let x ∈ T be of order 4. Then T = 〈x, q, c〉. By what we have shown above,
c γ = x 2 c and one of the following holds:
(*) xγ = x −1 and q γ ∈ 〈x 2 〉xq, or
(**) x γ = x and q γ ∈ 〈x 2 〉(xc)q.
We use now the well known fact that G ∼ = C2 × S4 has an outer automorphism µ of
order 2 such that c µ = c, x µ = xc and q µ = qc. If (**) holds then it is easy to check that
(*) holds for γ µ in place of γ. Also, if (αβ) µ = ch for some h ∈ T with h βµ
= h −1 ,
then αβ = ch µ and (hµ ) β = (h µ ) −1 . Similarly, if α µ β = ch such that h β = h −1 , then
α(β µ ) = ch µ and (hµ ) βµ
= (h µ ) −1 .
Thus, we may assume from now on that (*) holds for both α and β in place of γ.
After what we have just shown, it will be sufficient to prove αβ = ch for some h ∈ T
with h β = h −1 . We have [x, αβ] = [c, αβ] = 1 and q αβ ∈ 〈x 2 〉q. Therefore, αβ acts
on T like an element of 〈x〉. Since we suppose that (*) holds in place of β, we have in
particular that every element of 〈x〉 is inverted by β. This proves the assertion.
2.5 Linear groups
Throughout this section F is assumed to be a finite field of order q and V an n-
dimensional vector space over F for some n ≥ 1.
26

We write all vector spaces as right vector spaces.
By GL(V ) we denote the group of bijective linear transformations of V . We write
SL(V ) for the subgroup of GL(V ) consisting of all elements of GL(V ) with determi-
nant 1.
Note that, for given n and q, GL(V ) and SL(V ) are uniquely determined up to
isomorphism. Therefore we often write just GLn(q) or SLn(q) instead of GL(V ) or
SL(V ) respectively. Sometimes we will choose to write matrix representations for
elements of GL(V ). This corresponds to a particular choice of basis of V .
Definition 2.5.1. We set
P GL(V ) := GL(V )/Z(GL(V )).
and call P GL(V ) the projective linear group. The group
P SL(V ) := SL(V )/Z(SL(V ))
is called the special projective linear group and is sometimes also denoted by L(V ).
If n and q are given then the groups P GL(V ) and P SL(V ) are uniquely deter-
mined up to isomorphism. Therefore we will also write P GLn(q), P SLn(q) and Ln(q)
instead of P GL(V ), P SL(V ) and L(V ).
By Z we denote the subgroup of GL(V ) consisting of those linear transformation
which can be obtained by multiplication by an element of F ∗ . Then Z is isomorphic
to F ∗ . The following is well known.
Lemma 2.5.2. Z = CGL(V )(SL(V )) = Z(GL(V )) and Z(SL(V )) = Z ∩ SL(V ).
Notation 2.5.3. Let n ≥ 3. Then for each vector x = (x1, . . . , xn−1) ∈ F n−1 and each
27

matrix A ∈ GLn−1(q) we set
⎛
⎜
[x] = ⎜
⎝
In−1
0.
0
x1 . . . xn−1 1
⎞
⎟
⎠
⎛
⎜
and [A] = ⎜
⎝
A
0.
0
0 . . . 0 1
Lemma 2.5.4. Let n ≥ 2 and regard the elements of GLn(q) as matrices with entries
in F . Set GLn(q) := GLn(q)/Z, X := {[a] : a ∈ F n−1 } and E := {[A] : A ∈
GLn−1(q)}.
(a) X ∼ = X ∼ = F n−1 and [a][b] = [a + b] for all a, b ∈ F n−1 .
(b) E ∼ = E ∼ = GLn−1(q).
(c) [x] [A] = [xA] for all x ∈ F n−1 , A ∈ GLn−1(q).
(d) Let 0 �= x ∈ F n−1 and A ∈ GLn(q) such that [x] A
∈ X. Then A ∈ (E ⋉ X)Z.
(e) NGLn(q)(X) = (E ⋉ X)Z and NP GLn(q)(X) = E ⋉ X.
Proof. Claims (a), (b) and (c) are elementary to verify. Part (e) follows from (d). Thus
it is sufficient to show (d). For the proof set G = GLn(q) and V = F n . We consider
the natural action of G on V . Then there exist subspaces W, U of V such that W
has dimension n − 1, U has dimension 1, V = W ⊕ U, E = NG(W ) ∩ CG(U) and
X = CSLn(q)(W ).
It follows from the matrix representations that W = CV (X) = CV ([y]) for every
0 �= y ∈ F n−1 , and EXZ = NG(W ).
Now let 0 �= x ∈ F n−1 and A ∈ GLn(q) such that [x] A
⎞
⎟
⎠
∈ X. Then [x] A ∈ XZ.
Note that 1 ∈ F is the only Eigenvalue of [x], and [x] A has the same Eigenvalues
as [x]. Hence 1 is also the only Eigenvalue of [x] A . This yields [x] A ∈ X. Now
CV ([x]) = W = CV ([x] A ) and thus A ∈ NG(W ) = EXZ. Hence (d) holds.
28

2.6 2-Dimensional linear groups
In this section F will always be a finite field of order q and V a 2-dimensional vector
space over F . Note that by 2.5.2,
Z(SL(V )) = Z(GL(V )) ∩ SL(V ) = 〈z〉,
where z is the linear transformation corresponding to multiplication with −1 ∈ F .
In particular, Z(SL(V )) = 1 if p = 2, and |Z(SL(V ))| = 2 if p is odd.
Lemma 2.6.1. Let p �= 2. Then there is exactly one involution in SL(V ), namely the
one in Z(SL(V )).
Proof. This is [15, 8.6.2]
Set V # = V \{0} and for v ∈ V # let
vF := {vλ : λ ∈ F }
be the F -subspace of V generated by v. We set for v ∈ V # , P (v) = CSL(V )(v). If
v, w is a basis of V , then we may view P (v) as the group of matrices
�� 1 0
λ 1
�
�
: λ ∈ F ,
with respect to v, w. In particular, P (v) ∼ = (F, +) and thus P (v) ∈ Sylp(SL(V )) =
Sylp(GL(V )). Moreover, CV (P (v)) = vF = CV (x) for every 1 �= x ∈ P (v), and
NGL(V )(P (v)) = NGL(V )(vF ). Observe that we may view NGL(V )(P (v)) as the group
of matrices
�� δ1 0
λ δ2
�
: δ1, δ2 ∈ F ∗ �
, λ ∈ F ,
with respect to a basis v, w. In the next lemma we describe the Sylow structure of
SL(V ) and GL(V ).
29

Lemma 2.6.2. (a) Sylp(SL(V )) = Sylp(GL(V )) = {P (v) : v ∈ V # }. Moreover,
for u, v ∈ V # , we have
P (v) ∩ P (u) �= 1 ⇐⇒ P (v) = P (u) ⇐⇒ vF = uF.
(b) [V, P (v)] = vF and [V, P (v), P (v)] = 0 for v ∈ V # .
(c) |Sylp(GL(V ))| = q + 1.
(d) Sylp(GL(V )) = {P (u)} ∪ {P (v) x : x ∈ P (u)} for v, u ∈ V # with vF �= uF .
(e) SL(V ) acts 2-transitively on its Sylow p-subgroups.
Proof. See for example 8.6.4 and 8.6.5 in [15].
Lemma 2.6.3. Let T1, T2 be different Sylow p-subgroups of GL(V ). Then
Proof. See for example [15, 8.6.7].
SL(V ) = 〈T1, T2〉.
Lemma 2.6.4. SL2(2) is isomorphic to S3 and SL2(3) is a semidirect product of a
group of order 3 with a quaternion group of order 8 which is not direct.
Proof. See for example [15, 8.6.10].
Lemma 2.6.5. If q ≥ 4 then L(V ) is a non-abelian simple group.
Proof. See for example [15, 8.6.14].
Lemma 2.6.6. Φ(SL2(q)) = Z(SL2(q)).
Proof. Set G = SL2(q). Assume first that Z(G) �≤ Φ(G). Then p �= 2 and there
is a maximal subgroup H of G such that Z(G) �≤ H. Since |Z(G)| = 2, it follows
G = H × Z(G). Now |H| is even and therefore contains an involution. This is a
contradiction to 2.6.1. Hence, Z(G) ≤ Φ(G). Since by 2.6.4 and 2.6.5, G/Z(G) =
L2(q) is simple or isomorphic to S3 or S4, it follows Φ(G/Z(G)) = 1. Hence, Z(G) =
Φ(G).
30

By 2.5.2, every non-trivial element of P GL2(q) induces a non-trivial automor-
phism of SL2(q) and P SL2(q), i.e. P GL2(q) embeds into the automorphism group of
SL2(q) and P SL2(q).
Also note that Aut(F ) acts on GL2(q), SL2(q), P GL2(q) and P SL2(q). The ac-
tion of Aut(F ) on these groups is faithful. In particular, we can form the semidirect
product Aut(F ) ⋉ P GL2(q) and it embeds into Aut(SL2(q)).
Similarly, we can find an embedding of Aut(F ) ⋉ P GL2(q) into Aut(P SL2(q)).
By a classical result we have equality as described in the following theorem.
Theorem 2.6.7. Aut(SL2(q)) ∼ = Aut(P SL2(q)) ∼ = Aut(F ) ⋉ P GL2(q).
We remind the reader that for q = p n , Aut(F ) is cyclic of order n and generated
by the so called Frobenius automorphism of F which takes every element of F to its
p-th power.
The next three lemmas are also well known results and we state them without a
proof or reference.
Lemma 2.6.8. Let q be odd. Then P GL2(q) and P SL2(q) have dihedral Sylow 2-
subgroups.
Lemma 2.6.9. Let q be odd. Then there is a unique extension of P SL2(q 2 ) of degree
2 with semidihedral Sylow 2-subgroups.
Lemma 2.6.10. Let p �= 2. Then all involutions in L(V ) are conjugate in L(V ).
on.
We conclude this section with a rather specialized result which we will need later
Lemma 2.6.11. Let E be a normal subgroup of G such that G = ET and E ∼ = SL2(q)
for some power q of p. Assume that NG(T ∩ E) is p-closed. Then G = EOp(G).
31

Proof. If q ∈ {2, 3} then |T ∩ E| ∈ {2, 3}. Hence, T centralizes T ∩ E and by 2.6.7,
T ≤ (T ∩ E)CG(E), i.e. T = (T ∩ E)CT (E). Observe that CT (E) ≤ Op(G), so it
follows that G = ET = EOp(G).
Thus we may assume that q > 3. Then NE(T ∩ E) has order q(q − 1). Therefore,
by the Schur-Zassenhaus Theorem (see for example [15, 6.2.1]), there is a complement
K of T ∩ E in NE(T ∩ E). Moreover, all complements of T ∩ E in NE(T ∩ E) are
conjugate under NE(T ∩ E) and therefore under T ∩ E. Note that T acts on the set of
all complements of T ∩ E in NE(T ∩ E). Therefore, by the Frattini Argument 2.1.4,
T = NT (K)(T ∩ E). Hence, it is sufficient to show that NT (K) ≤ Op(G).
Set P = NT (K). Since we assume that NG(T ∩E) is p-closed, it follows [P, K] ≤
[T, K] ≤ T . Hence, [P, K] ≤ T ∩ K = 1. Note that K acts irreducibly on T ∩ E
and normalizes CT ∩E(P ) �= 1. Thus, [T ∩ E, P ] = 1 and P centralizes NE(T ∩ E).
Therefore, 2.6.7 implies P ≤ CT (E) ≤ Op(G). This shows that G = EOp(G).
2.7 Strongly p-embedded subgroups
In this section let H be a subgroup of G. Recall T ∈ Sylp(G).
Definition 2.7.1. The subgroup H is said to be strongly p-embedded in G if
• H �= G,
• p divides |H|, and
• for every x ∈ G\H, the order of H ∩ H x is not divisible by p.
Directly from the definition we get the following properties:
Remarks 2.7.2. Let H be strongly p-embedded, then the following hold:
(a) If P is a non-trivial p-subgroup of H, then NG(P ) ≤ H.
32

(b) Suppose E is a normal subgroup of G such that p divides |E|. Then E �≤ H.
Also, since P < NT (P ) for every proper subgroup P of T , 2.7.2(a) implies
Remark 2.7.3. If H is strongly p-embedded in G and T ∩ H �= 1, then T ≤ H. In
particular, every strongly p-embedded subgroup of G contains a Sylow p-subgroup of
G.
Together with 2.7.2 this implies
Remark 2.7.4. If G has a strongly p-embedded subgroup, then Op(G) = 1.
In view on 2.7.2(b), the normalizers of non-trivial p-subgroups of H play an im-
portant role in the understanding of strongly p-embedded subgroups. We define now
R(S, G) := 〈NG(P ) : 1 �= P ≤ S〉,
for every S ∈ Sylp(G). This leads to a new characterization of strongly p-embedded
subgroups, as stated in the following lemma.
Lemma 2.7.5. Assume H ∩ T �= 1. Then H is strongly p-embedded if and only if H is
a proper subgroup of G and R(T, G) ≤ H.
Proof. See [14, 17.11].
Corollary 2.7.6. The group G contains a strongly p-embedded subgroup if and only if
R(T, G) �= G.
Proof. If H is a strongly p-embedded subgroup of G, then by 2.7.3, T g ≤ H for
some g ∈ G. Thus, by 2.7.5, R(T g , G) �= G. Since R(T g , G) = R(T, G) g , it follows
R(T, G) �= G. On the other hand, if R(T, G) �= G, then by 2.7.5, R(T, G) is a strongly
p-embedded subgroup of G.
Example 2.7.7. Assume G ∼ = SL2(q) and H = NG(T ). Then H is a strongly p-
embedded subgroup of G.
33

Proof. Let V be a 2-dimensional vector space over GF (q) and suppose G = SL(V ).
Observe that H = NG(CV (T )) and CV (P ) = CV (T ) for 1 �= P ≤ T . Hence,
NG(P ) ≤ NG(CV (P )) = NG(CV (T )) = H. Now 2.7.5 yields the assertion.
Example 2.7.8. For n ≥ 4, Sn does not have a strongly 2-embedded subgroup.
Proof. For n = 4 this follows from 2.7.4. Let p = 2, G = Sn and H = R(T, G). By
2.7.5, it is sufficient to show that H = G. If n = 5 then T contains a fours group whose
normalizer is isomorphic to S4. So H �= G would imply H ∼ = S4. Then there would
be a transposition in G\H which centralizes a transposition in T , a contradiction to the
definition of H.
Thus, we may assume that n ≥ 6. Then n = 2m or n = 2m + 1 for some
natural number m ≥ 3. So there are m commuting transpositions in T . Then every
transposition in G is disjoint to at least one of these transpositions in T . Since G is
generated by its transpositions, it follows G = H.
Strongly p-embedded subgroups can often be inherited to subgroups and factor
groups as stated in the following two lemmas.
Lemma 2.7.9. Let T ≤ H ≤ G be strongly p-embedded.
(a) Let M be a subgroup of G which is not contained in H. If p divides the order of
H ∩ M, then H ∩ M is strongly p-embedded in M.
(b) Let N be a normal subgroup of G such that p divides |N|. Then H∩N is strongly
p-embedded in N.
Proof. Property (a) follows directly from the definition of strongly p-embedded sub-
groups. For the proof of (b) note that by 2.7.2(b), N �≤ H. Moreover, N∩T ∈ Sylp(N)
and hence p divides |N ∩ H|. Thus, (b) follows from (a).
34

Lemma 2.7.10. Let T ≤ H ≤ G be strongly p-embedded and N ✂ G such that
HN �= G. Then N is a p ′ group and HN/N is strongly p-embedded in G/N.
Proof. We have T ∩N ∈ Sylp(N) and by the Frattini Argument, G = NNG(T ∩N) �=
HN. Hence, NG(T ∩ N) �≤ H and therefore, since H is strongly p-embedded in G,
T ∩ N = 1. Hence, N is a p ′ -group.
We now set G = G/N. Let 1 �= P ≤ T . We need to show that N G (P ) ≤ H.
Let g ∈ G such that g ∈ N G (P ). Then P g N = (P N) g = P N. Since N is a p ′ -
group, P ∈ Sylp(NP ). Hence, there is h ∈ NP such that P gh = P . Then gh ∈ H,
since H is strongly p-embedded. Moreover, h −1 ∈ P N and P ≤ T ≤ H. Hence,
g = (gh)h −1 ∈ HN and g ∈ H.
Lemma 2.7.11. Assume G has a strongly p-embedded subgroup. Let N ✂ G be such
that N = E1 × . . . × Er for subgroups E1, . . . , Er of G. If p | |Ei| for i = 1, . . . , r,
then r = 1.
Proof. Let H ≤ G be strongly p-embedded such that T ≤ H. Since p divides |N|,
N �≤ H. Hence, there is i ∈ {1, . . . , r} such that Ei �≤ H. Assume r > 1. Then there
is i �= j ∈ {1, . . . , r}. Since Ej ✂ N and N ✂ G, Ej is subnormal in G. Therefore,
T ∩ Ej ∈ Sylp(G). In particular, T ∩ Ej �= 1. Since [Ei, Ej] = 1, it follows that
Ei ≤ NG(Ej ∩ T ) ≤ H, a contradiction.
Lemma 2.7.12. Let H ≤ G be strongly p-embedded in G and assume T ≤ H. Sup-
pose E is a normal subgroup of G such that E ∼ = SL2(q). Then H = NG(T ∩ E) and
G/E is a p ′ -group.
Proof. Set T0 = E ∩ T . The Frattini Argument gives us G = ENG(T0). By 2.7.2(a)
and 2.7.3, NG(T0) ≤ H. So it follows from Dedekind’s Law that H = NG(T0)(E∩H).
Assume E ∩ H �≤ NG(T0). Then there is T0 �= T1 ∈ Sylp(E ∩ H). Hence, 2.6.3 yields
E = 〈T0, T1〉 ≤ H. This is a contradiction since q divides |E|, E is normal in G and
35

H is strongly p-embedded in G. Thus, H = NG(T0).
Now let T0 �= T1 ∈ Sylp(E). Since H = NG(T0), there is g ∈ E\H such that
T g
0 = T1. Then NG(T0)∩NG(T1) = H ∩H g is a p ′ -subgroup of G, since H is strongly
p-embedded. Also, by 2.6.2, |Sylp(E)| = q + 1 and E acts 2-transitively on SylP (E).
Hence, G acts 2-transitively on Sylp(E) and therefore |G : NG(T0) ∩ NG(T1)| =
q(q + 1). Since we have shown above that NG(T0) ∩ NG(T1) is a p ′ -group, this yields
that q is the largest power of p dividing the order of G. So G/E is a p ′ -group.
2.8 Minimal parabolic subgroups
Definition 2.8.1. G is called minimal parabolic (with respect to p) if T is not normal
in G and there is a unique maximal subgroup of G containing T .
Note that the above definition does not depend on the choice of T , since the Sylow
p-subgroups of G are conjugate and every conjugate of a maximal subgroup of G is
again maximal in G.
A basic result about minimal parabolic subgroups is the following.
Lemma 2.8.2. Let G be minimal parabolic with respect to p, let M be the unique
maximal subgroup of G containing T , and let N be normal in G.
(a) If N ≤ M then N ∩ T ✂ G,
(b) If N �≤ M then O p (G) ≤ N.
Proof. By the Frattini Argument, we have G = NNG(N ∩T ). If N ≤ M, this implies
NG(T ∩ N) �≤ M. Since T ≤ NG(N ∩ T ) and M is the unique maximal subgroup of
G containing T , it follows that G = NG(T ∩ N). Hence (a) holds.
Assume now N �≤ M. If NT �= G, then by the choice of M, NT ≤ M and
hence N ≤ M, a contradiction. Thus G = NT and G/N is a p-group. This yields
O p (G) ≤ N.
36

The following lemma shows that, given a group G which is not p-closed, it is easy
to obtain minimal parabolic subgroups of G containing a Sylow p-subgroup of G.
Lemma 2.8.3. Let G0 be a subgroup of G such that NG(T ) ≤ G0 < G. Assume that
P is a subgroup of G which is minimal with the properties T ≤ P and P �≤ G0. Then
P is minimal parabolic and G0 ∩ P is the unique maximal subgroup of P containing
T .
Proof. We have T ∈ Sylp(P ). Since P �≤ G0 and NG(T ) ≤ G0, T is not normal in
P . Now let M be a maximal subgroup of P containing T , and assume M �= G0 ∩ P .
Then, since M is maximal, M �≤ G0 ∩ P . Hence, M �≤ G0. Also, T ≤ M < P , a
contradiction to the minimality of P .
2.9 Uniqueness of amalgams
Throughout this section, A = (G1, G2, G12, φ1, φ2) and B = ( ˜ G1, ˜ G2, ˜ G12, ψ1, ψ2) are
amalgams, as defined in the introduction.
Definition 2.9.1. A triple (β1, β2, β) of group isomorphisms β : G12 → ˜
G12 and
βi : Gi → ˜ Gi, for i = 1, 2, is said to be an isomorphism from A to B, if the obvious
diagram commutes, i.e. if φiβi = βψi, for i = 1, 2.
Recall that G1 ∗G12 G2 denotes the free product of G1 and G2 with G12 amal-
gamated. Here we suppress mention of φ1, φ2 and thus of the exact isomorphism
type of A. However, the amalgam should always be clear from the context. Usually
we identify G1, G2 and G12 with their images in G1 ∗G12 G2, so the monomorphisms
φi : G12 → Gi become inclusion maps which we, by abuse of notation, denote by id.
Also, if H is a group and φ : G12 → H is a group isomorphism, then A is isomorphic
to (G1, G2, H, φ −1 φ1, φ −1 φ2) via (id, id, φ). Thus, if we want to show that A and B
are isomorphic, we usually may assume that G12 = ˜ G12, and that G12 is contained in
G1, G2, ˜ G1 and ˜ G2. The next remark deals with this situation.
37

Remark 2.9.2. Suppose H := G12 = ˜ G12 and H is contained G1, ˜ G1, G2 and ˜ G2.
Assume A = (G1, G2, H, id, id) and B = ( ˜ G1, ˜ G2, H, id, id). Then A and B are
isomorphic if and only if there exist group isomorphisms α1 : G1 → ˜ G1 and α2 :
G2 → ˜ G2 such that α1|H = α2|H.
Lemma 2.9.3. Assume G1 is a dihedral or semidihedral 2-group of order at least 16,
G2 ∼ = S4 and G12 ∼ = D8. Then, up to isomorphism, the amalgam A is uniquely
determined by the isomorphism type of G1.
Proof. Assume G12 ∼ = ˜ G12 and Gi ∼ = ˜ Gi for i = 1, 2. We will show that A and B
are isomorphic. As argued above, we may assume that G12 = ˜ G12, that H := G12 is
contained in G1, G2, ˜ G1, ˜ G2, and that φ1, φ2, ψ1, ψ2 are inclusion maps.
Since H is a Sylow 2-subgroup of G2 and G2 ∼ = ˜ G2, there exists an isomorphism
α2 : G2 → ˜ G2 such that H α2 = H. Let α be the automorphism of H induced by α2.
Since G1 and ˜ G1 are isomorphic, it follows from 2.4.3(g) that there is an isomorphism
β : G1 → ˜ G1 such that H β = H. Then (β|H) −1 α ∈ Aut(H), and by 2.4.3(h), there
exists an inner automorphism γ of ˜ G1 such that γ|H = (β|H) −1 α. Now set α1 := βγ.
Then (α1, α2, α) is an isomorphism from A to B.
Lemma 2.9.4. Let H ≤ G and let σ, τ ∈ Aut(H) be involutions. For h ∈ H let ch be
the inner automorphism of H determined by h. Suppose there exists α ∈ Aut(G) such
that H α = H and one of the following holds:
(*) α −1 σαInn(H) = τInn(H) and Z(H) = 1, or
(**) α −1 σα = τch for some h ∈ H with h τ = h −1 .
Then the amalgam (〈σ〉 ⋉ H, G, H, id, id) is isomorphic to (〈τ〉 ⋉ H, G, H, id, id).
Proof. We will identify H and τ with there images in 〈τ〉 ⋉ H. Recall that then the
image of an element h ∈ H under τ is just the same as h conjugate by τ inside 〈τ〉⋉H.
38

Therefore, it will be consistent to denote both by h τ . Similarly we identify H and σ
with there images in 〈σ〉 ⋉ H.
If (*) holds then σ α = τch for some h ∈ H and Z(H) = 1. It follows o(τch) =
o(σ) = 2. This gives us ˆ h = ˆ(τch) 2
h
= ˆ h τhτh for every ˆ h ∈ H. Thus, h τ h = τhτh ∈
Z(H) = 1. Therefore, h τ = h −1 . So we may assume from now on that (**) holds. By
2.9.2, it is sufficient to show the following.
(+) There exists a group isomorphism α ∗ : 〈σ〉H → 〈τ〉H such that α ∗ |H = α|H.
By (**), we have o(τh) = 2 inside of 〈τ〉H. This yields 〈τ〉H = 〈τh〉 ⋉ H. Since
τh acts on H by conjugation in the same way as the automorphism τch acts, it follows
from 2.1.2 that there exists a group isomorphism β : 〈τch〉H → 〈τ〉H such that
β|H = id and (τch)β = τh. Thus, for the proof of (+), we can replace τ by τch and
may assume that σ α = τ. In this case, (+) follows from 2.1.3.
Lemma 2.9.5. Let G ∼ = C2 × S4 and T ∈ Syl2(G). Suppose S is a group containing
T as a subgroup of index 2. If Op(G) and Z(G) are not normal in S, then the amalgam
(S, G, T, id, id) is uniquely determined up to isomorphism.
Proof. Set Q = Op(G) and C = Z(G). Note that S = S/Z(T ) is not abelian,
since Q is not normal in S. Moreover, S has order 8 and T is elementary abelian of
order 4. Hence, S is dihedral and there exists u ∈ S\T such that u is an involution.
Then X := Z(T )〈u〉 has order 8 and is non-abelian since, by assumption, [C, u] �= 1.
Moreover, Z(T ) is a fours group. Hence, X is dihedral and there is an involution
t ∈ X\Z(T ). Note that X ∩ T = Z(T ) and thus t �∈ T . So we have S = 〈t〉 ⋉ T .
Now by 2.4.6 and 2.9.4, A is uniquely determined up to isomorphism.
Lemma 2.9.6. Let F = GF (4), Q = F 2 , M = SL2(4) and G = M ⋉ Q, where we
consider the natural action of M on Q. Let T ∈ Syl2(G), H = NG(T ) and let X be a
2-closed group containing H as a subgroup of index 2. Assume Q is not normal in X.
Then the amalgam (X, G, H, id, id) is uniquely determined up to isomorphism.
39

Proof. Let A be the group consisting of the elements of Aut(H) which extend to an
automorphism of G. Fix D ∈ Syl3(H) and let ˆ D be the image of D in Inn(H) ∼ = H.
Set C = CAut(H)(D) and B = NC(Q).
For S ∈ Syl2(X) we have |S/T | = 2 and D acts on S. Observe that D acts fixed
point freely on T . Thus, S = CS(D) ⋉ T and X = CS(D) ⋉ H. Moreover, the
non-trivial element of CS(D) induces an automorphism σ of H of order 2 that does
not normalize Q. By 2.1.2 and 2.9.2, the amalgam (X, G, H, id, id) is isomorphic to
the amalgam (〈σ〉⋉H, G, H, id, id). Note that this argument works for every choice of
X. Also, Z(H) = 1. So by 2.9.4, it is sufficient to show the following for C = C/ ˆ D.
(+) The involutions in C\B are conjugate under A ∩ C.
Note that there is exactly one elementary abelian subgroup R of T of order 2 4 such
that R �= Q. Then D acts on R, and C acts on {Q, R}. In particular, |C : B| ≤ 2.
Therefore, in order to show (+), it is sufficient to show the following.
(++) The involutions in Bτ are conjugate under A ∩ C for some involution τ ∈ C\B.
For V ∈ {Q, R} let ΩV be the set of D-invariant complements of Z := Z(T ) in V .
Note that |ΩQ| = |ΩR| = 4 and C acts on ΩQ ∪ ΩR. We show next:
(E1) There is ˆ D ≤ D1 ≤ A ∩ B and τ ∈ C\B such that D1 ∼ = C3 × C3, o(τ) = 2,
and D1 is inverted by τ. Moreover, D1 acts non-trivially on ΩV , for every V ∈
{Q, R}.
For the proof of (E1) we identify G with a subgroup of P SL3(4) containing the image
of the lower triangular matrices as described in 2.5.4. Set
⎛ ⎞ ⎛
1 0 0
0 0 1
D∗ := ⎝ 0 λ 0 ⎠ , M := ⎝ 0 1 0
0 0 1
1 0 0
and SL3(4) := SL3(4)/Z(SL3(4)). Let τ0 be the automorphism of P SL3(4) which is
the contragredient automorphism followed by the automorphism induced by conjuga-
tion with M. Then it is easy to check that D∗ is inverted by τ0. Also, we can choose
40
⎞
⎠

D ∈ Syl3(G) such that D is centralized by τ0. Moreover, D∗ and τ0 normalize G.
Now we can define τ to be the restriction of τ0 to an automorphism of H, and D1 to be
the image of D〈D∗〉 in Aut(H). Then (E1) follows.
(E2) There is E ≤ M ∩ B such that E ∼ = C2 × C2 and [Q, E] = 1.
For the proof of (E2) we use that there exists an M-module W over GF (2) that is
an extension of Q such that |W/Q| = 4 and CW (M) = 1. We consider G then as a
subgroup of the semidirect product MW . Note that M acts trivially on W/Q. Thus,
G is normal in MW and Q = [W, M]. Then W0 := QCW (T ) is an M-module and
|W0/CW0(T )| = |Q/CQ(T )| = 4. Since M is generated by two conjugates of T ∩ E
and CW (M) = 1, it follows |W0| ≤ 4 2 . Thus, W0 = Q and CW (T ) = CQ(T ).
Therefore, every element of CW (D) acts non-trivially on H and we can define E to be
the image of CW (D) in Aut(H). This shows (E2).
Now let P ∈ Syl3(B). Then for Y := CP (Q/Z) ∩ CP (R/Z), [Q, Y, R] = 1 =
[R, Y, Q]. Hence, by the Three-Subgroup Lemma, [Z, Y ] = [Q, R, Y ] = 1. Since the
action of Y on T is coprime, it follows [T, Y ] = 1 = [H, Y ] and Y = 1. Using (E1)
we get |P/CP (R/Z)| = 3 = |P/CP (Q/Z)| and thus,
P = CP (R/Z) × CP (Q/Z) ∼ = C3 × C3.
Since |C : B| = 2, every element of C of odd order is contained in B. A non-trivial
element of B of odd order has to act non-trivially on Z, Q/Z or R/Z. Therefore, 2
and 3 are the only prime divisors of |C|. In particular, for K ∈ Syl2(B), B = P K.
Observe that CK(R) ∩ CK(Q) = 1. Also, every non-trivial element of K/CK(R)
acts fixed point freely on ΩR, since D acts irreducibly on each member of ΩR. Thus,
|K/CK(R)| ≤ 4. Now (E2) yields K/CK(R) ∼ = C2 × C2. By the Frattini Argument,
we may conjugate with an element from NC(K)\B and get K/CK(R) ∼ = C2 × C2.
Hence, K has order 2 4 and CK(V ) acts fixed point freely on ΩU for {U, V } = {Q, R}.
41

Observe that, in particular, no element in C\ ˆ D acts non-trivially on ΩQ∪ΩR. It follows
now from (E1) that there exists a monomorphism φ : X → S8 such that
τφ = (15)(26)(37)(48),
CK(Q) = Op(S4),
Kφ = Op(S4) × Op(S4) τφ ,
D1φ = 〈(123)(578)〉.
Then it is easy to check that there are exactly eight 3-elements in B = KD1 that are
inverted by τ. Also, there are exactly three involutions in B that are centralized by τ.
Hence, there are precisely 12 involutions in Bτ.
For L = 〈D1, E〉 with D1 and E as in (E1) and (E2), we have |L| = 12. Note
that E = CK(Q) and L ≤ A. By (E1), τ inverts B/O2(B) and therefore does not
centralize any 3-element in L. Hence, C L (τ) ≤ O2((L)) = E. Now the identification
via φ above gives us C L (τ) = 1. Since there are exactly |L| = 12 involutions in Bτ,
this shows that L acts transitively on the involutions in Bτ. Because L ≤ A, this
implies (++). As we have argued above, this proves the assertion.
42

Chapter 3
Groups Acting on Modules
Throughout this chapter let G be a group, p a prime and T ∈ Sylp(G).Furthermore, let
F be a finite field of characteristic p and V a finite dimensional F G-module.
3.1 Preliminaries
Notation 3.1.1. Write End F (V ) for the set of F -linear transformations from V to
itself. Set
End F G(V ) := {φ ∈ End F (V ) : (vφ) g = (v g )φ for every g ∈ G}.
If this does not lead to any confusion we will write EndG(V ) instead of EndF G (V ).
If V is irreducible, i.e. if 0 and V are the only G-invariant F -subspaces of V , then
by Schur’s Lemma, EndF G (V ) is a skew field. Moreover, by Wedderburn’s Theorem,
every finite skew field is a field. Thus, if V is irreducible, K := EndF G (V ) is a finite
field and V is an KG-module.
Also note that multiplication by an element of F is an F -endomorphism of V which
commutes with every element of G. Therefore F is isomorphically contained in K and
can be regarded as a subfield of K.
Lemma 3.1.2. Let A and B be finite irreducible F G-modules and let φ : A → B be
an F G-module isomorphism. Set FA := EndF G (A) and FB := EndF G (B). Then φ is
43

semilinear with respect to FA and FB, i.e. there exists a field isomorphism σ : FA →
FB such that (aλ)φ = (aφ)λ σ for all a ∈ A, λ ∈ FA.
Proof. For λ ∈ FA set σλ = φ −1 λφ. Since φ and λ are F G-module homomorphisms,
σλ is an F G-module homomorphism as well, i.e. σλ ∈ FB. Therefore, we can define
a map σ : FA → FB by λ ↦→ σλ. Then σ is a field isomorphism and (aλ)φ = (aφ)λ σ
for all a ∈ A, λ ∈ FA.
3.2 Offender
The basic definition of this section is the following:
Definition 3.2.1. • A subgroup A of G is said to be an offender on V , if
(a) A/CA(V ) is a non-trivial elementary abelian p-group,
(b) |V/CV (A)| ≤ |A/CA(V )|.
• The module V is called an FF-module for G, if there is an offender in G on V .
If V is an FF-module, it is in many situations possible to obtain strong information
about the structure of G and the action of G on V . We will come to an example of that
in Section 3.5.
FF-modules usually arise in situations where V is an elementary abelian normal
p-subgroup of the acting group G. We will explain next how in such situations V can
often be shown to be an FF-module.
Definition 3.2.2. We write A(G) for the set of all elementary abelian p-subgroups of
G of maximal order. The group generated by A(G) is denoted by J(G) and called the
Thompson subgroup of G.
Lemma 3.2.3. Let V be an elementary abelian normal p-subgroup of G. Let A ∈
A(G) and suppose that A does not centralize V .
44

(a) A is an offender on V .
(b) If |V/CV (A)| ≥ |A/CA(V )|, then V CA(V ) ∈ A(G). In particular, we have
A(CG(V )) ⊆ A(G) and J(CG(V )) ≤ J(G).
Proof. Since V CA(V ) is an elementary abelian subgroup of T , we have |V CA(V )| ≤
|A|. Hence,
|V/CV (A)| ≤ |V/V ∩ A| = |V/V ∩ CA(V )| = |V CA(V )/CA(V )| ≤ |A/CA(V )|
Therefore A is an offender on V . If |V/CV (A)| = |A/CA(V )|, then we have equality
above and in particular, |V CA(V )| = |A|. Hence (b) holds.
Lemma 3.2.3 also shows that in the case |V/CV (A)| = |A/CA(V )| we are often able
to obtain some extra information. This motivates the following definition.
Definition 3.2.4. An offender A on V is called an over-offender on V , if we have
|V/CV (A)| < |A/CA(V )|.
Lemma 3.2.5. Let V, W be normal elementary abelian p-subgroups of G with V ≤ W
and [V, J(G)] �= 1. Let A ∈ A(T ) such that [V, A] �= 1, and ACG(W ) is a minimal
with respect to inclusion element of the set
{BCG(W ) : B ∈ J(G), [W, B] �= 1}.
Assume A is not an over-offender on V . Then we have |W/CW (A)| = |A/CA(W )| =
|V/CV (A)| and W = V CW (A).
Proof. It follows from 3.2.3 that |V/CV (A)| = |A/CA(V )|, B := CA(V )V ∈ A(T )
and |W/CW (A)| ≤ |A/CA(W )|. Since [V, A] �= 1 and [V, B] = 1, BCG(W ) is a
proper subset of ACG(W ). Hence, the minimality of ACG(W ) yields [B, W ] = 1.
Thus CA(V ) = CA(W ). It follows that
|W/CW (A)| ≤ |A/CA(W )| = |A/CA(V )|
= |V/CV (A)| = |V CW (A)/CW (A)| ≤ |W/CW (A)|.
45

Now equality holds above, i.e |A/CA(W )| = |W/CW (A)| = |V/CV (A)| and W =
V CW (A).
Lemma 3.2.6. Let V be an elementary abelian normal p-subgroup of G such that
[V, J(G)] �= 1. Set G = G/CG(V ) and assume there is no over-offender in G on V .
Then there is A ∈ A(G) such that A is an offender on V which is minimal with respect
to inclusion.
Proof. Let A ∈ A(G) such that [V, A] �= 1. Choose A so that ACG(V ) is minimal
with respect to inclusion. By 3.2.3, A is an offender on V . Let B ≤ A such that B
is an offender on V . We may assume that CA(V ) ≤ B. Then CA(V ) = CB(V ) and
A ∩ V = B ∩ V . Since there is no over-offender in G on V , we have
|A/CA(V )||CV (A)| = |V | = |B/CB(V )||CV (B)|.
Thus, |A||CV (A)| = |B||CV (B)| and therefore,
|A| = |ACV (A)| = |A||CV (A)||A ∩ V | −1 = |B||CV (B)||B ∩ V | −1 = |BCV (B)|.
This yields BCV (B) ∈ A(G). Now by the choice of A, BCG(V ) = ACG(V ). Thus,
A = B and the assertion holds.
In view of 3.2.3(b), 3.2.5 and 3.2.6 we will later be particularly interested in showing
that, on certain modules, over-offenders do not exist.
Lemma 3.2.7. Assume W is a G-submodule of V such that G acts faithfully on V and
V = V/W . Then the following hold:
(a) Every offender on V is an offender on V .
(b) If there is no over-offender in G on V , then there is no over-offender in G on V .
Proof. Let A be an offender on V . Then
|V /C V (A)| ≤ |V /CV (A)| ≤ |V/CV (A)| ≤ |A/CA(V )| = |A| = |A/CA(V )|.
46

If there is no over-offender in G on V , then we have equality above and A is not an
over-offender on V .
The concept of an offender as introduced above has the drawback that an offender on
a given module is not necessarily an offender on any submodule. However, it turns out
that every submodule of a given FF-module is again an FF-module. Moreover, there is
a special type of offenders which can be inherited to submodules.
Definition 3.2.8. We call a subgroup A of G a best offender on V if
• A/CA(V ) is non-trivial and elementary abelian,
• |A/CA(V )||CV (A)| ≥ |A ∗ /CA ∗(V )||CV (A ∗ )| for all subgroups A ∗ of A.
We write OG(V ) for the set of all best offenders in G on V and JG(V ) for the subgroup
generated by all best offenders in G on V .
Remark 3.2.9. Every best offender on V is an offender on V .
Proof. Let A ≤ G be a best offender on V . Then we have for A ∗ = 1, |V | =
|A ∗ /CA ∗(V )||CV (A ∗ )| ≤ |A/CA(V )||CV (A)|. Hence, A is an offender on V .
Not every offender is a best offender. However, we have
Lemma 3.2.10. The module V is an FF-module iff there is a best offender on V .
Proof. By 3.2.9, it is sufficient to show that there exists a best offender on V if V
is an FF-module. Let A be an offender on V . Choose B to be a subgroup of A
with [V, B] �= 1 for which |B/CB(V )||CV (B)| is maximal. Then by this choice,
|B/CB(V )||CV (B)| ≥ |B ∗ /CB ∗(V )||CV (B ∗ )| for every subgroup B ∗ of B for which
[B ∗ , V ] �= 1. Now let B ∗ be a subgroup of B centralizing V . Again by the maximality
of |B/CB(V )||CV (B)|,
|B/CB(V )||CV (B)| ≥ |A/CA(V )||CV (A)| ≥ |V | = |B ∗ /CB ∗(V )||CV (B ∗ )|.
Hence B is a best offender.
47

We will use 3.2.9 and 3.2.10 most of time without reference. As we required, best
offenders can be inherited to submodules. More precisely:
Lemma 3.2.11. Let W be a G-submodule of V . Then {A ∈ OG(V ) : [W, A] �= 1} ⊆
OG(W ).
Proof. Let A ∈ OG(V ) and A ∗ ≤ A. Set B = A ∗ CA(W ). Then CW (B)CV (A) ≤
CV (B). Hence, |CV (B)| ≥ |CW (B)| |CV (A)| |CW (A)| −1 . Since A ∈ OG(V ), it
follows
|A/CA(V )| |CV (A)| ≥ |B/CB(V )| |CV (B)| ≥ |B/CB(V )| |CW (B)| |CV (A)| |CW (A)| −1 .
This implies together with CA(V ) = CB(V ) that
Since CW (B) = CW (A ∗ ), this gives us
|A| |CW (A)| ≥ |B| |CW (B)|.
|A/CA(W )| |CW (A)| ≥ |B/CA(W )| |CW (B)| = |A ∗ /CA ∗(W )| |CW (A ∗ )|.
The converse of the last lemma is not true, but at least we have the following.
Lemma 3.2.12. Let W be a G-submodule of V and A ∈ OG(W ) such that V =
W CV (A). Then A ∈ OG(V ).
Proof. Because V = W CV (A), we have for A ∗ ≤ A that CV (A ∗ ) = CV (A)CW (A ∗ )
and CA ∗(V ) = CA ∗(W ). Using this and A ∈ OG(W ) we get
|CV (A ∗ )||A ∗ /CA∗(V )| = |CV (A)||CW (A)| −1 |CW (A ∗ )||A ∗ /CA∗(W )|
≤ |CV (A)||CW (A)| −1 |CW (A)||A/CA(W )|
= |CV (A)||A/CA(V )|.
48

We obtain examples of best offenders in a similar way to which we obtained examples
of offenders.
Lemma 3.2.13. Let V be a normal subgroup of G and A ∈ A(G) such that [V, A] �= 1.
Then A ∈ OG(V ).
Proof. See [8, 2.8(e)].
Definition 3.2.14. A subgroup A of G is called a quadratic best offender on V if A is
a best offender on V and [V, A, A] = 1.
Lemma 3.2.15 (Timmesfeld Replacement Theorem I). Every best offender on V con-
tains a subgroup which is a quadratic best offender.
Proof. This is [15, 9.2.3].
The next lemma is a version of the Timmesfeld Replace Theorem for elementary
abelian subgroups of maximal order.
Lemma 3.2.16 (Timmesfeld Replacement Theorem II). Let G be a p-group and V an
elementary abelian normal subgroup of G. Let A ∈ A(G) such that [V, A] �= 1. Then
there exists B ∈ A(ACG(V )) such that [V, B] �= 1 and [V, B, B] = 1.
Proof. Note that by 3.2.13, A ∈ OG(V ). Hence by [15, 9.2.1],
|A/CA(V )| |CV (A)| = |A ∗ /CA ∗(V )| |CV (A ∗ )| for A ∗ = CA([V, A]).
Also, CA ∗(V ) = CA(V ) and hence,
|A| = |A ∗ | · |CV (A ∗ )|
|CV (A)| .
Since V ∩ A ∗ ≤ CV (A), it follows now for B := A ∗ CV (A ∗ ) that
|B| = |A∗ | |CV (A ∗ )|
|V ∩ A ∗ |
≥ |A|.
Note also that B is elementary abelian. Hence, B ∈ A(ACG(V )). By the definition of
B, [V, B, B] = 1. It follows from [15, 9.2.3] that [V, B] = [V, A ∗ ] �= 1.
49

3.3 Natural Sn-modules
Definition 3.3.1. If G ∼ = Sm for some m ≥ 3, then we call a GF (2)-module a per-
mutation module for G, if it has a basis {v1, v2, . . . , vm} of length m on which G acts
faithfully.
A GF (2)-module is called a natural Sm-module if it is isomorphic to a non-central
irreducible section of the permutation module.
Natural Sm-modules are by this definition uniquely determined up to isomorphism.
Suppose V is a permutation module for Sm with basis {v1, v2, . . . , vm} as above. Set
W = 〈vi + vj : 1 ≤ i, j ≤ m〉 and U = 〈v1 + v2 . . . + vm〉. If m is odd, then
W ∼ = V/U and the natural module is isomorphic to both W and V/U. In particular, a
natural Sm-module has dimension m − 1. If m is even, then U ≤ W and the natural
Sm-module is isomorphic to W/U. Accordingly, it has dimension m − 2.
Example 3.3.2. Assume p = 2, G = S2n+1 and V is a natural G-module. Then the
following conditions hold:
(a) The elements in OG(V ) are precisely the subgroups generated by commuting
transpositions.
(b) NG(JT (V ))/JT (V ) ∼ = Sn.
(c) If n > 2 and J is a subgroup of T generated by elements of OT (V ), then NG(J)
is not p-closed.
Proof. Part (a) follows from [8, 2.15]. Note that T contains precisely n transposi-
tions t1, . . . , tn, which pairwise commute. By (a), JT (V ) is generated by t1, . . . , tn.
Furthermore, NG(JT (V )) acts on {t1, . . . , tn} and CG(t1, . . . , tn) = JT (V ). Hence
NG(JT (V ))/JT (V ) is isomorphic to a subgroup of Sn.
50

If (ij) and (kl) are distinct transpositions in JT (V ), then conjugation with the el-
ement (ik)(jl) ∈ NG(JT (V )) swaps (ij) and (kl). Hence, NG(JT (V ))/JT (V ) ∼ = Sn
since Sn is generated by transpositions. This shows (b).
Suppose now n > 2 and let J be a subgroup of T generated by elements of OT (V ).
If J = JT (V ), then (c) follows from (b). Thus we may assume that J �= JT (V ). This
means that J is generated by less than n transpositions and CG(J) contains a subgroup
isomorphic to S3. Since S3 is not p-closed, this shows (c).
3.4 Natural SL2(q)-modules
The group SL2(q) acts, by definition, faithfully on a 2-dimensional vector space over
GF (q), i.e. such a vector space provides a GF (q)-module for SL2(q). We first observe
that a natural SL2(q)-module as defined in the introduction has exactly the structure of
such a module.
Remark 3.4.1. Let G ∼ = SL2(q) and V be a natural SL2(q)-module for G. Set
F := End
GF (p)
G
(V ). Let χ : G → GL2(F ) be the group homomorphism mapping
each element g ∈ G to the F -linear transformation of V induced by g. Then χ is a
monomorphism and Gχ = SL2(F ).
Lemma 3.4.2. Assume that G ∼ = SL2(q) and V is a natural SL2(q)-module for G.
Then
(a) |CV (T )| = q,
(b) CV (T ) = [V, T ] = CV (a) for each a ∈ T # ,
(c) T acts quadratically on V ,
(d) CG(CV (T )) = T .
Proof. This follows from 2.6.2 and 3.4.1.
51

Lemma 3.4.3. Let G ∼ = SL2(q) and V be a natural SL2(q)-module for G. Then
OG(V ) = {A ≤ G : A is an offender on V } = Sylp(G). Moreover there are no
over-offenders in G on V .
Proof. By 3.4.2(a), T ∈ Sylp(G) is an offender on V . If B is an offender on V , then
CV (B) is a non-trivial proper subspace of V . Hence q = |V/CV (B)| ≤ |B| ≤ q, i.e.
B ∈ Sylp(G) and B is not an over-offender. Now the assertion follows from 3.2.9 and
3.2.10.
We now apply the results from above to situations in which natural SL2(q)-modules
occur inside groups.
Hypothesis 3.4.4. Set Q = Op(G) and assume
• Q is elementary abelian,
• G/Q ∼ = SL2(q),
• Q/CQ(G) is a natural SL2(q)-module for G/Q.
Lemma 3.4.5. Assume Hypothesis 3.4.4.
(a) Q ∈ A(T ).
(b) For A ∈ A(T )\{Q}, T = AQ, Z(T ) = CQ(A) = A ∩ Q = CQ(t) for every
t ∈ T \Q and |Q/Z(T )| = q.
Proof. Assume there is A ∈ A(T ) such that A �= Q. Then |A| ≥ |Q| and A �≤ Q.
Therefore by 3.4.2, |Q/CQ(A)| ≥ q. In particular,
q ≤ |Q/CQ(A)| ≤ |Q/Q ∩ A| ≤ |A/Q ∩ A| = |AQ/Q| ≤ |T/Q| = q.
Then equality holds above. So |Q| = |A| and (a) holds. Moreover, T = QA, |Q/(Q ∩
A)| = q and CQ(A) = Q ∩ A ≤ Z(T ). This, together with 3.4.2(b), implies (b).
52

Lemma 3.4.6. Assume Hypothesis 3.4.4 and p = 2. Then |A(T )| ≤ 2. Furthermore,
if |A(T )| = 2, then every involution in T is contained in an element of A(T ).
Proof. We may assume that there exists A ∈ A(T ) such that A �= Q. Then by 3.4.5(b),
T = QA. Thus, if t ∈ T \Q is an involution, then there exist a ∈ A\Q and x ∈ Q
such that t = ax. Since t, a and x are involutions, then a and x commute. Therefore
by 3.4.5(b), x ∈ CQ(a) ≤ Z(T ) = Q ∩ A. Hence, t ∈ A.
So we have shown that Q # ∪A # is the set of involutions in T and therefore A(T ) =
{Q, A}. This proves the assertion.
Lemma 3.4.7. Assume Hypothesis 3.4.4 and CQ(G) = 1. Let d ∈ NG(T ) be of order
q − 1 and x ∈ Aut(T ) such that T = QQ x and (t d ) x = (t x ) d for all t ∈ T . Then
q ∈ {2, 4}.
Proof. Set F := EndG(Q), Z := Z(T ) and D = 〈d〉. Let v1, v2 be an F -basis of Q
such that v1 ∈ Z(T ) and there is λ ∈ F with v d 1 = v1λ and v d 2 = v2λ −1 . From now
on we write Q as an additive group. Set A := Q/Z and B := Q x /Z. Then D acts
irreducibly on A and B. Therefore, by Schur’s Lemma and Wedderburn’s Theorem,
FA := EndD(A) and FB := EndD(B) are fields. Furthermore, |FA| ≤ |A| = q and
|FB| ≤ |B| = q. Also note that every element of D can be regarded as an element of
F ∗ A or F ∗ B . Hence,
FA ∼ = FB ∼ = F ∼ = GF (q).
Every element of F acts on the factor space A and can be regarded as an element of
FA. From now on we identify F with FA in this way and get
(3.1)
Note that by 2.2.2,
a d = aλ −1 for all a ∈ A.
φ : B → Z, defined by tZ ↦→ [v2, t], for all t ∈ Q x ,
53

is a group isomorphism. For µ ∈ F = End
GF (p)
G
(Q) we now set ψµ = φµφ −1 . Then
ψµ is the composition of group homomorphisms and therefore also a group homomor-
phism from B to itself. Hence we may regard ψµ as an element of End GF (p) (B). It is
easy to check that ψ : F → End GF (p) (B) is an isomorphism of rings. Thus we can
identify F with its image in End GF (p) (B). In particular, we set
We show next that
(3.2)
bµ := bψµ for all b ∈ B and µ ∈ F.
b d = b(λ 2 ) for all b ∈ B.
For the proof let b = tZ ∈ B for some t ∈ Q x . Note that [v2λ, t] = [v2, t]λ since
(v2λ) t = v t 2λ. Hence, b d φ = [v2, t d ] = [v d−1
2 , t] d = [v2λ, t]λ = [v2, t]λ 2 = (bφ)λ 2 .
Thus, b d = (b d φ)φ −1 = (bφ)λ 2 φ −1 = bψ λ 2 = bλ 2 and (3.2) holds. In particular,
(b d )µ = (bµ) d for all b ∈ B, µ ∈ F . Therefore, via the identification above, F = FB.
Now x induces a group homomorphism A → B, and thus a GF (p)-linear trans-
formation. Since by assumption, (a d ) x = (a x ) d for every a ∈ A, it follows from 3.1.2
that there is a field isomorphism
σ : FA → FB
such that (aµ) x = (a x )µ σ for all a ∈ A, µ ∈ FA. If we identify FA and FB with F as
above, then we can regard σ as a field automorphism of F . Now it follows from (3.1)
and (3.2) that
(a x )(λ −1 ) σ = (aλ −1 ) x = (a d ) x = (a x ) d = (a x )λ 2 for all a ∈ A.
Hence, (λ−1 ) σ = λ2 . Note that there is e ∈ N such that pe ≤ q and µ σ = µ pe for all
µ ∈ F . Then 1 = λ 2 λ σ = λ 2 λ pe
shows q ∈ {2, 4}.
= λ pe +2 , i.e. q − 1 divides p e + 2 ≤ q + 2. This
54

3.5 Recognizing SL2(q)
In this section we want to summarize and apply some results of Bundy, Hebbinghaus
and Stellmacher [8].
Notation 3.5.1. Suppose that OG(V ) �= ∅. Set
mG(V ) := max{|A/CA(V )||CV (A)| : A ∈ OG(V )},
and define AG(V ) to be the set of minimal by inclusion elements of the set
{A ∈ OG(V ) : |A/CA(V )||CV (A)| = mG(V )}.
Note that mG(V ) ≥ |V |, since every element of OG(V ) is an offender.
Definition 3.5.2. Let D be a set of subsets of G. Then D is called normal (in G) if
A g ∈ D for all A ∈ D, g ∈ G. Moreover, we define
For U ≤ G we set
NG(D) = {g ∈ G | A g ∈ D for all A ∈ D }.
D ∩ U := {A : A ∈ D, A ≤ U} and DG(U) := {A ∈ D : A ≤ U g for all g ∈ G}.
Hypothesis 3.5.3. Suppose that G acts faithfully on V , and D is a normal set of non-
trivial elementary abelian subgroups of G such that the following hold for A ∈ D:
(i) [V, A, A] = 1.
(ii) |V/CV (A)| = |A| and CV (A) = CV (a) for every a ∈ A # .
(iii) |V/CV (AB)| = |A||B| for every B ∈ D with A �= B and [A, B] = 1.
(iv) |A||CV (A)| ≥ |X||CV (X)| for every elementary abelian p-subgroup X ≤ G.
(v) There is T ∈ Sylp(G) and T ≤ M ≤ G with D �= DG(M) such that
NG(D ∩ T ) ≤ M and CG(CV (T )) ≤ M.
55

Theorem 3.5.4. Assume Hypothesis 3.5.3. Then there exist subgroups E1, . . . , Er of
G such that the following hold for Wi := [V, Ei] and i, j ∈ {1, . . . , r}:
(a) D = DG(M) ∪ (D ∩ E1) ∪ . . . ∪ (D ∩ Er) and 〈D〉 = 〈DG(M)〉 × E1 × . . . × Er.
(b) V = WiCV (Ei) and if i �= j then [Wi, Ej] = [Wi, 〈DG(M)〉] = 1.
(c) Ei ∼ = SL2(qi) where qi = |A| for A ∈ D ∩ Ei, or Ei ∼ = Sm, m odd, Ei ∩ M ∼ =
Sm−1, and |A| = 2 for A ∈ D ∩ Ei.
(d) Ei ∼ = SL2(qi) and Wi/CWi (Ei) is a natural SL2(qi)-module for Ei, or Ei ∼ = Sm
and Wi is a natural Sm-module for Ei. Moreover, in the second case D ∩ Ei acts
as the conjugacy class of transpositions on Wi.
Proof. This is Theorem 4.18 in [8].
Theorem 3.5.5. Suppose G acts faithfully on V and G is minimal parabolic with re-
spect to p. Let T ∈ Sylp(G) and M ≤ G be the unique maximal subgroup of G
containing T . Assume also that
(i) Op(G) = 1,
(ii) V is an FF-module for G,
(iii) CG(CV (T )) ≤ M.
Then for D = AG(V ), there exist subgroups E1, . . . , Er of G such that for each 1 ≤
i ≤ r the following hold:
(a) P = (E1 × . . . Er)T .
(b) T acts transitively on {E1, . . . , Er}.
(c) D = (D ∩ E1) ∪ . . . ∪ (D ∩ Er).
(d) V = CV (E1 × . . . × Er) � r
i=1 [V, Ei], with [V, Ei, Ej] = 1 for j �= i.
56

(e) Ei ∼ = SL2(p n ), or p = 2 and Ei ∼ = S2 n +1 for some n ∈ N.
(f) [V, Ei]/C[V,Ei](Ei) is a natural module for Ei.
Proof. This is Theorem 5.5 in [8].
Lemma 3.5.6. Assume G acts faithfully on V and Op(G) = 1. Let p = 2, T ∈ Syl2(G)
and let E1, . . . , Er be subgroups of G. Set Wi = [V, Ei] for 1 ≤ i ≤ r. Suppose that
for i = 1, . . . , r the following conditions hold:
(a) G = (E1 × . . . × Er)T and T acts on {E1, . . . , Er}.
(b) V = CV (E1 × . . . × Er) � r
i=1 Wi and [Wi, Ej] = 1 for i �= j.
(c) There is n ∈ N such that Ei ∼ = S2n+1 and Wi/CWi (Ei) is a natural S2n+1-
Then
module for Ei.
JT (V ) = 〈JT ∩E1(W1), . . . , JT ∩Er(Wr)〉.
Proof. For B ∈ OT ∩Ei (Wi) it follows from (b) that V = CV (B)Wi. Hence, 3.2.12
implies B ≤ JT (V ).
Now let A ∈ OT (V ) and set Vi = [V, Ei]. By [8, 2.16(a)], A ≤ NG(O 2 (Ei)) for
i = 1, . . . , r. Set E = E1 × . . . × Er and H = E �
i NG(O 2 (Ei)). Then A ≤ H,
H is normal in G and O 2 (Ei) is normal in H for i = 1, . . . , r. Since O 2 (Ei) ∼ =
A2n+1, Aut(O 2 (Ei)) ∼ = S2n+1 ∼ = Ei and H = EiCH(O 2 (Ei)) for 1 ≤ i ≤ r. It
follows inductively that CH(O 2 (E1), . . . , O 2 (El)) = El+1CH(O 2 (E1), . . . , O 2 (El+1))
and H = E1 . . . El+1CH(O 2 (E1) . . . O 2 (El+1)) for all 0 ≤ l < r. Hence, H =
E1 . . . ErCH(O 2 (E1) . . . O 2 (Er)) = ECH(O 2 (E)). We have E ∩ CH(O 2 (E)) = 1
and CH(O 2 (E)) ∼ = H/E is a 2-group, since G/E is a 2-group. Note that CH(O 2 (E))
is normal in H and O2(H) is normal in G. So it follows CH(O 2 (E)) ≤ O2(H) ≤
57

O2(G) = 1. Hence, H = E and A ≤ E.
For i = 1, . . . r define a projection map
πi : A → T ∩ Ei , a1a2 . . . ar ↦→ ai for all a1 ∈ E1, . . . , ar ∈ Er with a1 . . . ar ∈ A.
Set Ai = Aπi for i = 1, . . . , r. Let i ≤ r and let  be a subgroup of Ai. Set
A ∗ = π −1
i (Â). Then N := Kerπi acts trivially on Vi, A/N ∼ = Ai and [Vi, a] = 1 if
and only if [Vi, aπi] = 1 for each a ∈ A. Hence, |Ai/CAi (Vi)| = |A/N/CA/N(Vi)| =
|A/CA(Vi)|. Similarly, | Â/CÂ (Vi)| = |A∗ /CA∗(Vi)|. Moreover, CVi (A) = CVi (Ai)
and CVi (Â) = CVi (A∗ ). Assume Ai �= 1. Then [Vi, A] �= 1. Hence, by 3.2.11,
A ∈ OE(Vi). Therefore,
|Ai/CAi (Vi)| |CVi (Ai)| = |A/CA(Vi)| |CVi (A)|
Thus, Ai ∈ OT ∩Ei (Vi).
Theorem 3.5.7. Suppose
• G acts faithfully on V ,
• G is minimal parabolic,
• V is an FF-module for G, and
• G has a strongly p-embedded subgroup.
≥ |A ∗ /CA ∗(Vi)| |CVi (A∗ )| = | Â/C Â (Vi) |CVi (Â)|.
Then for q = |T |, G ∼ = SL2(q) and V/CV (G) is a natural SL2(q)-module for G.
In particular, we have
(a) OG(V ) = Sylp(G) and |T ||CV (T )| = |V |,
(b) [V, T, T ] = 1,
58

(c) CG(CV (T )) ≤ G0, and
(d) CV (a) = CV (T ) for a ∈ T # .
Proof. Let T ∈ Sylp(G) and let G ≤ G0 ≤ G be a maximal strongly p-embedded
subgroup. Then G0 is the unique maximal subgroup of G containing T . Set
Z0 = CV (O p (G)) and V = V/Z0.
Since G has a strongly p-embedded subgroup, we have
(3.3)
Op(G) = 1.
For C := CG(V ) we have O p (C) ≤ O p (G). Thus, [V, O p (C)] = [V, O p (C), O p (C)] ≤
[Z0, O p (C)] = 1. Hence O p (C) = 1, since G acts faithfully on V . Then C is a normal
p-subgroup and by (3.3), C = 1. Therefore, we have
(3.4)
It follows now from 3.2.7 that
(3.5)
We show next that
(3.6)
G acts faithfully on V .
OG(V ) �= ∅.
CG(C V (T )) ≤ G0.
Let Z0 ≤ Z1 ≤ V such that Z1 = C V (T ). Assume CG(Z1) �≤ G0. Recall that we have
chosen G0 such that G0 is the unique maximal subgroup of G containing T . Hence,
T ≤ CG(Z1) implies CG(Z1) = G. Then [Z1, O p (G), O p (G)] ≤ [Z0, O p (G)] = 1.
Thus, C V (T ) = Z1 = 1. It follows V = 1, a contradiction, since G acts faithfully on
V . Therefore (3.6) holds.
Now by (3.3), (3.4), (3.5) and (3.6), we can apply 3.5.5 with V instead of V .
Together with 2.7.11, 2.7.9(b) and 2.7.8, this gives us G = ET for some normal
59

subgroup E of G with E ∼ = SL2(q). Moreover, V /C V (E) is a natural SL2(q)-module
for E. Hence, by 2.7.12, G = E. Together with C V (E) = 1 this implies
(3.7)
G ∼ = SL2(q), q = |T | and V is a natural SL2(q) − module.
Now 3.2.7 and 3.4.3 imply OG(V ) = OG(V ) = Sylp(G) and |V/CV (T )| = |T | = q.
Thus, (a) holds.
By 2.6.3, G is generated by two conjugates of T , hence |V/CV (G)| ≤ q 2 = |V |
and V = V/CV (G).
Property (b) follows from 3.2.15. Hence, we get together with 3.4.2 that CV (T ) ≤
C V (T ) = [V , T ] ≤ CV (T ), i.e. C V (T ) = CV (T ). Thus, (3.6) implies (c).
For a ∈ T # we have, again by 3.4.2, CV (a) ≤ C V (a) = C V (T ) = CV (T ), i.e.
CV (a) ≤ CV (T ). Hence, (d) holds and the proof is complete.
Theorem 3.5.8. Assume
• G acts faithfully on V ,
• G = Op′ (G),
• V is an FF-module for G, and
• G has a strongly p-embedded subgroup.
Then G ∼ = SL2(q) and V/CV (G) is a natural SL2(q)-module for G.
Proof. Let T ∈ Sylp(G) and let G0 be a strongly p-embedded subgroup of G contain-
ing T . Let T ≤ P ≤ G such that P is minimal with the property P �≤ G0. Then
by 2.8.3, P is minimal parabolic and G0 ∩ P is the unique maximal subgroup of P
containing T . Moreover, by 2.7.9, G0 ∩ P is strongly p-embedded in P . Note also that
P acts faithfully on V since G acts faithfully on V . Also, V is an FF-module for P
because of OT (V ) �= ∅.
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Hence, we can apply Theorem 3.5.7 to P . This gives us
(1) [V, T, T ] = 1,
(2) |T ||CV (T )| = |V |,
(3) CV (a) = CV (T ) for a ∈ T # ,
(4) CP (CV (T )) ≤ G0.
If we assume CG(CV (T )) �≤ G0, then we can choose P such that P ≤ CG(CV (T ))
and get a contradiction to (4). Therefore,
(4’) CG(CV (T )) ≤ G0.
Set D = Sylp(G). We want to verify Hypothesis 3.5.3. Properties (i), (ii) and (iv) of
3.5.3 follow from (1), (2) and (3).
If S ∈ D such that [S, T ] = 1, then ST is a p-subgroup of G and hence T = ST =
S. Thus, (iii) of Hypothesis 3.5.3 holds.
Since G0 is strongly p-embedded in G, DG(G0) = ∅. Moreover, NG(D ∩ T ) =
NG(T ) ≤ G0. Now (4’) implies (v) of 3.5.3. Hence, Hypothesis 3.5.3 holds and
we may apply Theorem 3.5.4. This gives us together with 2.7.11, 2.7.9(b) and 2.7.8
that G = Op′ (G) = 〈D〉 ∼ = SL2(q) for q = |T |, and V/CV (G) is a natural SL2(q)-
module.
61

Chapter 4
Pushing Up
Let H be a finite group, p a prime dividing |H| and T ∈ Sylp(H). Let q be a power of
p.
4.1 Applying a classical result
H is said to have the pushing up property (with respect to p) if the following holds:
(PU) No non-trivial characteristic subgroup of T is normal in H.
Note that this property does not depend on the choice of T since all Sylow p-subgroups
of H are conjugate in H. The problem of determining the non-central chief factors of
H in Op(H) under the additional hypothesis
(*) H/Φ(H) ∼ = L2(q) for H = H/Op(H)
was first solved by Baumann in [5] and by Niles in [17]. Later Stellmacher gave in
[21] a shorter proof.
Theorem 4.1.1. Suppose H fulfils (PU) and (*), and T is not elementary abelian. Set
V = [Op(H), O p (H)]. Then there exists x ∈ Aut(T ) such that
L/V0Op ′(L) ∼ = SL2(q) for L = V x O p (H) and V0 = V (L ∩ Z(H)),
and one of the following holds:
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(I) V ≤ Z(Op(H)) and V/CV (H) is a natural SL2(q)-module for L/V0Op ′(L).
(II) Z(V ) ≤ Z(Op(H)), p = 3, and Φ(V ) = CV (H) has order q. Moreover,
V/Z(V ) and Z(V )/Φ(V ) are natural SL2(q)-modules for L/V0Op ′(L).
Furthermore, the following hold:
(a) Op(H) = V COp(H)(L).
(b) If (II) holds then Op(H) x2 = Op(H) for every x ∈ Aut(T ) with V x �≤ Op(H).
(c) Φ(COp(H)(O p (H))) x = Φ(COp(H)(O p (H))) for every x ∈ Aut(T ) with V x �≤
Op(H).
Proof. By Theorem 1 in [21] and [17, 3.2], (I) or (II) hold. Property (a) is a part of
Theorem 2 in [21]. Parts (b) and (c) follow from 2.4, 3.2, 3.3 and 3.4(c) in [21].
In order to apply this Theorem in our situation, we determine the structure of a finite
group G fulfilling the following hypothesis:
Hypothesis 4.1.2. Let G be a finite group, T ∈ Sylp(G) and let W ≤ Ω(Z(Op(G)))
be normal in G. Suppose the following conditions hold:
(1) G/CG(W ) ∼ = SL2(q),
(2) W/CW (G) is a natural SL2(q)-module for G/CG(W ),
(3) CG(W )/Op(G) is a p ′ -group,
(4) G �= CG(W )NG(C) for every characteristic subgroup C of T .
Theorem 4.1.3. Suppose Hypothesis 4.1.2 holds. Then there is a subgroup H of G
with T ≤ H and G = CG(W )H such that for V := [Op(G), O p (H)] one of the
following holds:
(I) V ≤ W and V/CV (G) is a natural SL2(q)-module for H/CH(W ).
63

(II) Z(V ) ≤ Z(Op(G)), p = 3, and Φ(V ) = CV (H) has order q. Moreover,
V/Z(V ) and Z(V )/Φ(V ) are natural SL2(q)-modules for H/CH(W ).
Furthermore the following hold:
(a) Op(G) = V COp(G)(L) for some subgroup L of H with H = LOp(G).
(b) If (II) holds, then Op(G) x2 = Op(G) for every x ∈ Aut(T ) with V x �≤ Op(G).
(c) Φ(COp(G)(O p (H))) x = Φ(COp(G)(O p (H))) for each x ∈ Aut(T ) with V x �≤
Op(G).
Proof. Let T ≤ H ≤ G be minimal such that G = CG(W )H. Then by (4) of 4.1.2,
H has the pushing up property. Also, H/CH(W ) ∼ = SL2(q) and Op(G) = Op(H).
Set G = G/Op(G). We want to show (*). Assume CH(W ) �≤ Φ(H). Then there
is a maximal subgroup H1 of H with Op(G) ≤ H1 such that CH(W ) �≤ H1. The
maximality of H1 yields H = H1CH(W ) and G = CG(W )H = CG(W )H1. Since
CH(W )/Op(H) is a p ′ -group, H1 contains a Sylow p-subgroup of H. Hence, after
conjugation with an element of H, we may assume that T ≤ H1. Now the minimal
choice of H yields H = H1, a contradiction. Therefore CH(W ) ≤ Φ(H).
Using 2.6.6 we get
H/Φ(H) ∼ = H/CH(W )/Φ(H/CH(W )) ∼ = L2(q).
Thus, H fullfills the hypothesis of Theorem 4.1.1. Hence there exists x ∈ Aut(T )
such that
L/V0Op ′(L) ∼ = SL2(q) for L = V x O p (H) and V0 = V (L ∩ Z(H)),
and (I) or (II) of Theorem 4.1.1 hold. Moreover (b) and (c) follow immediately from
(b) and (c) of Theorem 4.1.1. Note that, in particular, Z(V ) = [W, O p (H)]CV (H)
and CL(W ) ≤ CL(Z(V )/CV (H)) = Op ′(L)V0. Moreover, V � H and V ≤ L.
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Thus, [V, Op ′(L)] = V ∩ Op ′(L) = 1. Since W = [W, Op (H)]CW (H) ≤ V CW (H),
also [W, Op ′(L)] = 1. Hence, CL(W ) = Op ′(L)V0 ≤ CL(V )V . In particular,
LCH(W ) = H and LOp(G) contains a Sylow p-subgroup of H. This implies to-
gether with O p (H) ≤ L that H = LOp(G). Now (a) is a consequence of Theorem
4.1.1(a).
It remains to show that H/CH(W ) acts on V/CV (H). Since H = LOp(G), it fol-
lows from (a) and the above that CH(W ) = Op(G)CL(W ) ≤ CL(V )V . Note also that
4.1.1 shows in particular that [V, V ] ≤ CV (H). Hence, [V, CH(W )] ≤ CH(W ). This
completes the proof of 4.1.3.
4.2 The Baumann subgroup
A useful subgroup while dealing with pushing up situations is the following:
Definition 4.2.1. Let P be a p-group then
is called the Baumann subgroup of P .
B(P ) = CP (Ω(Z(J(P ))))
Often it is not possible to show immediately that H has the pushing up property. In
many of these situations it helps to look at a subgroup X of H such that B(T ) ∈
Sylp(X) and to show that X has the pushing up property. Here one uses that a char-
acteristic subgroup of B(T ) is also a characteristic subgroup of T . Usually one can
then restrict the order of X and thus also of B(T ) and Ω(Z(J(T ))). This often leads
to T = B(T ) ≤ X, in which case also the p-structure of H is restricted.
When using this method later, we will need the following lemma:
Lemma 4.2.2. Let V be a normal elementary abelian p-subgroup of H such that
• H/CH(V ) ∼ = SL2(q),
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• V/CV (H) is a natural SL2(q)-module for H/CH(V ),
• CH(V )/Op(H) is a p ′ -group and [V, J(T )] �= 1.
Then there is a subgroup X of H such that H = CH(V )X and B(T ) ∈ Sylp(X).
Proof. Set H = H/CH(V ) and let A ∈ A(T ) such that [V, A] �= 1. Then by 3.4.3
and 3.2.3, |V/CV (A)| = |A| = q and V ≤ J(T ). Therefore, T = J(T )Op(H) and,
since CH(V )/Op(H) is a p ′ -group, Ω(Z(J(T ))) ≤ CT (V ) = Op(H). Furthermore,
W := Ω(Z(J(T )))V is elementary abelian and
|W/CW (J(T ))| = |V CW (J(T ))/CW (J(T ))| = |V/CV (J(T ))| = |V/CV (T )| = q.
Now let d ∈ H such that for X0 = 〈J(T ), J(T ) d 〉, H = X0CH(V ). Then
|V CW (X0)/CW (X0)| ≤ |W/CW (X0)| ≤ q 2 = |V/CV (X0)| = |V CW (X0)/CW (X0)|.
Hence we have equality above and therefore W = V CW (X0).
Set Z0 = CΩ(Z(J(T )))(X0). Since V ≤ J(T ), we have W ≤ J(T ) and CW (X0) ≤
Ω(Z(J(T ))). Thus CW (X0) = Z0 and W = V Z0. Now Dedekind’s Law implies
Ω(Z(J(T ))) = Z0(Ω(Z(J(T ))) ∩ V ). So, using T = Op(H)J(T ), we get B(T ) =
CT (Z0).
Note that Z0 = Ω(Z(J(T ))) ∩ Ω(Z(J(T d ))) and therefore Z0 is normal in M :=
〈T, T d 〉 = 〈J(T ), J(T ) d 〉Op(H). Hence, X := CM(Z0) ✂ M. Since T ∈ Sylp(M),
this yields B(T ) = T ∩ X ∈ Sylp(X). Moreover H = CH(V )X0 = CH(V )X.
66

Chapter 5
Preliminaries on Fusion Systems
5.1 Basic definitions
Let G be a group. For g ∈ G we write cg : G → G for the inner automorphism of G
determined by g.
Let P and Q be subgroups of G. Set
NG(P, Q) = {g ∈ G | P g ≤ Q}.
For any map φ : P → Q, A ≤ P and Aφ ≤ B ≤ G we denote by φ|A,B the map with
domain A and range B mapping each element of A to its image under φ. In particular
for x ∈ NG(P, Q), cx|P,Q is the restriction of cx to the domain P and the range Q. Set
MorG(P, Q) = {cg |P,Q | g ∈ NG(P, Q)}.
For P ≤ Q ≤ G, ιP,Q denotes the natural embedding of P into Q, i.e. the map from P
to Q which maps each element of P to itself.
Definition 5.1.1. Let S be a group. A fusion system on S is a category F whose
objects are all subgroups of S and whose morphisms satisfy the following properties
for all P, Q ≤ S.
(1) MorF(P, Q) is a set of injective group homomorphisms, containing MorS(P, Q).
(2) The composition of morphisms in F is the same as the composition of group
homomorphisms.
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(3) For each φ ∈ MorF(P, Q), φ|P,P φ ∈ MorF(P, P φ).
(4) If φ ∈ MorF(P, Q) is surjective, then the inverse map φ −1 is an element of
MorF(Q, P ).
The main class of examples for fusion systems is the following.
Example 5.1.2. Let S be a subgroup of G. Write FS(G) for the category whose objects
are all the subgroups of S, and for objects P, Q ∈ FS(G),
Then FS(G) is a fusion system on S.
MorFS(G)(P, Q) = MorG(P, Q).
From now on let S be a group and F a fusion system on S.
By an abuse of notation we will write F for the set of all objects of F. In particular
we write Q ∈ F instead of Q ≤ S.
Note that by axiom (4), an F-morphism is an isomorphism in the sense of category
theory if and only if it is a group isomorphism, and the inverse map is then also the
inverse in the categorical sense. Moreover we can think of the inclusion maps between
appropriate subgroups of S as maps obtained by conjugation with 1 ∈ S. Hence, by
the first axiom, they all are morphisms in F. Also note that by axiom (3) we can factor
every F-morphism as an F-isomorphism followed by inclusion. Thus, usually it is
sufficient to consider properties of F-isomorphisms.
Assume now P, Q ∈ F and φ ∈ MorF(P, Q) is an isomorphism. Let A ≤ P
and Aφ ≤ B ≤ S. Then φ|A,Q = ιA,P φ ∈ MorF(A, Q). Therefore by axiom (3)
applied to φ|A,Q instead of φ, φ|A,Aφ ∈ MorF(A, Aφ). Hence φ|A,B = φ|A,AφιAφ,B ∈
MorF(A, B). So, for a given F-morphism, we can restrict its source and take any
suitable target, and again obtain an F-morphism.
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Definition 5.1.3. A subsystem of F is a fusion system E on a subgroup T of S such
that for all A, B ≤ T , MorE(A, B) ⊆ MorF(A, B).
Assume now we are given a family (Fi : i ∈ I) of fusion systems Fi on subgroups
Si of S. The fusion system E is called the intersection of all Fi, if E is a fusion system
on T := �
i∈I Si, and MorE(A, B) = �
MorFi i∈I (A, B) for all A, B ≤ T .
The fusion systems F is said to be generated by (Fi : i ∈ I), if F is the intersection
of all those fusion systems on S which contain each Fi as a subsystem. We then write
F = 〈Fi : i ∈ I〉.
Note here that the intersection of fusion systems as defined above is indeed again a
fusion system. Also, if we are given a group S, then we can form the fusion systems
on S whose morphisms are all injective group homomorphisms between subgroups of
S. This fusion system contains every fusion system on S. Therefore, the generation
of fusion systems as above is well defined. Moreover, if F = 〈Fi : i ∈ I〉 for a
family of fusion systems (Fi : i ∈ I), then the morphisms in F are precisely the group
homomorphisms between subgroups of S which are the composition of morphisms
from the Fi.
Definition 5.1.4. Let ˜ F be a fusion system on a group ˜ S. Then an isomorphism of
groups α : S → ˜ S is called an isomorphism from F to ˜ F if
Mor ˜ F (P α, Qα) = {α −1 φα : φ ∈ MorF(P, Q)} for all P, Q ∈ F.
The fusion systems F and ˜ F are called isomorphic if there exists an isomorphism
between them.
If ˜ F is a fusion system as above and α an isomorphism between F and ˜ F, then for
P, Q ∈ F the maps
αP,Q : MorF(P, Q) → MorF(P α, Qα), φ ↦→ α −1 φα
69

are bijective. Moreover, together with the map F → ˜ F defined by P ↦→ P α, they
give an isomorphism from the category F to the category ˜ F. Thus, if F and ˜ F are
isomorphic, then they are also isomorphic as categories.
Example 5.1.5. Let G, H be groups, S ≤ G and let φ : G → H be an isomorphism of
groups. Then the map from S to Sφ induced by φ is an isomorphism of fusion systems
from FS(G) to FSφ(H).
Definition 5.1.6. Let P, Q ∈ F.
• We say P is fused into Q if MorF(P, Q) �= ∅.
• Q is said to be F-conjugate to P if there exists an isomorphism in MorF(P, Q).
By P F we denote the F-conjugacy class of P , i.e. the set of all subgroups of S
which are F-conjugate to P .
Note that for two F-conjugate subgroups P, Q of S all the elements of MorF(P, Q)
are isomorphisms.
Definition 5.1.7. Let P ∈ F.
• The subgroup P is called centric if for every Q ∈ P F , CS(Q) ≤ Q.
• Define P to be fully centralized (fully normalized) if for all Q ∈ P F , |CS(P )| ≥
|CS(Q)| (|NS(P )| ≥ |NS(Q)| respectively).
Remark 5.1.8. Assume P is fully centralized and CS(P ) ≤ P . Then P is centric.
Proof. Let Q ∈ P F . Since P is fully centralized, |Z(Q)| ≤ |CS(Q)| ≤ |CS(P )| =
|Z(P )| = |Z(Q)|. Thus, CS(Q) = Z(Q) ≤ Q.
Notation 5.1.9. • For P ∈ F we set
AutF(P ) = MorF(P, P ).
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Similarly, we set for P ≤ R ≤ S,
AutR(P ) = MorR(P, P ).
• If P, Q ∈ F and φ ∈ MorF(P, Q) is an isomorphism, then we write φ ∗ for the
map
φ ∗ : AutF(P ) → AutF(Q) defined by ψ ↦→ φ −1 ψφ.
Note that for every P ∈ F, AutF(P ) is a group acting on P . Also, AutR(P ) is a
subgroup of AutF(P ) for each R ∈ F containing P . Furthermore, observe that for
all P, Q ∈ F and every isomorphism φ ∈ MorF(P, Q), the map φ ∗ : AutF(P ) →
AutF(Q) is an isomorphism of groups.
Remark 5.1.10. Let P, R, T ∈ F, φ ∈ MorF(R, T ) and assume P ≤ R ≤ NS(P ).
Then
cg |P,P (φ|P,P φ) ∗ = cgφ |P φ,P φ for every g ∈ R.
Notation 5.1.11. Let P, Q ∈ F and φ ∈ MorF(P, Q) be an isomorphism. Set
Nφ = {g ∈ NS(P ) | cg |P,P φ ∗ ∈ AutS(Q)}.
Remark 5.1.12. Let P, Q ∈ F and φ ∈ MorF(P, Q) be an isomorphism. Then
CS(P )P ≤ Nφ.
Proof. By 5.1.10 we have cg |P,P φ ∗ = cgφ |Q,Q ∈ AutS(Q) for all g ∈ P . Thus P ≤ Nφ.
Moreover for g ∈ CG(P ), cg |P,P φ ∗ = idP φ ∗ = idQ = c1|Q,Q ∈ AutS(Q).
We are now able to define saturated fusion systems.
Definition 5.1.13. F is said to be saturated if S is a finite p-group and the following
conditions hold for every P ∈ F:
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(I) If P is fully normalized, then P is fully centralized and we have AutS(P ) ∈
Sylp(AutF(P )).
(II) If Q ∈ P F is fully centralized and φ ∈ MorF(P, Q), then φ extends to a member
of MorF(Nφ, S).
Example 5.1.14. Let G be a finite group, S ∈ Sylp(G) and F = FS(G). Then F is a
saturated fusion system on S.
Moreover the following hold for every P ∈ F:
(a) P is fully centralized if and only if CS(P ) ∈ Sylp(CG(P )).
(b) P is fully normalized if and only if NS(P ) ∈ Sylp(NG(P )).
Proof. See for example [16, 2.10 and 2.11].
5.2 Saturated fusion systems
For the remainder of this chapter let F be a saturated fusion system on S. Moreover
we set
for every P ∈ F.
A(P ) = AutF(P )
Remark 5.2.1. Let P, Q ≤ S such that Q is fully centralized, and let φ ∈ MorF(P, Q)
be an isomorphism. Then φ extends to a member of MorF(CS(P )P, CS(Q)Q).
Proof. Since by 5.1.12, CS(P )P ≤ Nφ this follows from axiom (II) of Definition
5.1.13.
Notation 5.2.2. Let Q, R ∈ F such that R ≤ NS(Q). Then we set
RQ = AutR(Q).
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Lemma 5.2.3. Let Q, R ∈ F such that QCS(Q) ≤ R ≤ NS(Q) and Q is fully cen-
tralized. Then φ ∈ A(Q) extends to some element of A(R) if and only if φ normalizes
RQ.
Proof. Suppose first φ ∈ A(Q) normalizes RQ. Then RQφ ∗ = RQ ≤ AutS(Q) and
hence R ≤ Nφ. By axiom (II) of Definition 5.1.13, φ extends to an element ˆ φ ∈
MorF(Nφ, S). Then 5.1.10 implies that c g ˆ φ|Q,Q ∈ RQ for g ∈ R. Since CS(Q) ≤ R,
this shows R ˆ φ = R and φ extends to an element of A(R). The other direction follows
from 5.1.10.
Lemma 5.2.4. Let Q ∈ F.
(a) Q is fully centralized if and only if for each P ∈ Q F there exists a morphism
φ ∈ MorF(CS(P )P, CS(Q)Q) such that P φ = Q.
(b) Q is fully normalized if and only if for each P ∈ Q F there exists a morphism
φ ∈ MorF(NS(P ), NS(Q)) such that P φ = Q.
Proof. Assume for P ∈ Q F there exists a morphism φ ∈ MorF(CS(P )P, CS(Q)Q)
such that P φ = Q. Then CS(P )φ = CS(Q). Since all morphisms are injective,
|CS(Q)| ≥ |CS(P )|. The other direction of (a) follows from 5.2.1.
If for all P ∈ Q F , MorF(NS(P ), NS(Q)) �= ∅ then clearly Q is fully normal-
ized. Assume now that Q is fully normalized and let P ∈ Q F . Then there exists an
isomorphism α ∈ MorF(P, Q) and AutS(P )α ∗ is a p-subgroup of A(Q). Since F is
saturated and Q fully normalized, AutS(Q) ∈ Sylp(A(Q)). Hence, by Sylow’s Theo-
rem, there exists β ∈ A(Q) such that AutS(P )(αβ) ∗ = (AutS(P )α ∗ )β ∗ ≤ AutS(Q).
Then NS(P ) = Nαβ. Therefore, since Q is fully normalized and F saturated, αβ ex-
tends to some φ ∈ MorF(NS(P ), NS(Q)). Then P φ = P αβ = Qβ = Q. This shows
(b).
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5.3 Normalizers
Definition 5.3.1. Let P ∈ F.
• We define NF(P ) (the normalizer in F of P ) to be the category whose objects
are the subgroups of NS(P ) such that for A, B ≤ NS(P ), MorNF (P )(A, B) is
the set of all φ ∈ MorF(A, B) which extend to an element of MorF(AP, BP )
taking P to P .
• The subgroup P is called normal in F if F = NF(P ); that is P ✂ S and for
all R, Q ≤ S, each φ ∈ MorF(R, Q) extends to a member of MorF(RP, QP )
which normalizes P . We write P ✂ F to indicate that P is normal in F.
• By Op(F) we denote the largest subgroup of S which is normal in F
Note here that NF(P ) is a fusion system on NS(P ).Moreover Op(F) is well defined
since the product of two normal subgroups of F is again normal in F.
Proposition 5.3.2. Let P be fully normalized in F. Then NF(P ) is a saturated fusion
system on NS(P ).
Proof. This is a direct consequence of Proposition A.6 in [7].
Remark 5.3.3. Let P ∈ F and P ≤ Q ≤ NS(P ). Then AutNF (P )(Q) = NA(Q)(P ).
Lemma 5.3.4. Let G be a finite group, S ∈ Sylp(G) and F = FS(G). Let P ≤ S be
fully normalized in S with respect to F. Then NS(P ) ∈ Sylp(NG(P )) and NF(P ) =
FNS(P )(NG(P )).
Proof. By 5.1.14(b), we have NS(P ) ∈ Sylp(NG(P )). Note that both NF(P ) and
FNS(P )(NG(P )) are fusion systems on NS(P ).
Let A, B ≤ NS(P ). If φ ∈ MorNF (P )(A, B) then there exists a morphism ˆ φ ∈
MorF(AP, BP ) such that P ˆ φ = P and ˆ φ|A,B = φ. Since F = FS(G), there exists
74

g ∈ G such that ˆ φ = cg |AP,BP . Then φ = ˆ φ|A,B = cg |A,B and P g = P ˆ φ = P , i.e.
g ∈ NG(P ) and φ is a morphism in FNS(P )(NG(P )).
Now let ψ = cg |A,B for some g ∈ NG(P ) ∩ NG(A, B). Then ˆ ψ = cg |AP,BP is an
extension of ψ such that P ˆ ψ = P g = P . Hence ψ is a morphism in NF(P ).
Lemma 5.3.5. Let G be a finite group, S ∈ Sylp(G) and P ≤ S. If P ✂ G then
P ✂ FS(G).
Proof. This follows from 5.3.4.
As the following example shows the converse of Lemma 5.3.5 is wrong in general.
Example 5.3.6. Set G = SL2(q) for some power q of p, S ∈ Sylp(G) and F = FS(G).
Then by 2.7.7, H = NG(S) is a strongly p-embedded subgroup of G. Therefore F =
FS(H) and 5.3.5 implies that S is normal in F. However, S is not normal in G.
5.4 Factor systems
Definition and remark 5.4.1. Let R ≤ S be a normal subgroup of F. Since R ✂ S
we may set S = S/R. For P, Q ≤ S and φ ∈ MorF(P R, QR) we can define a group
isomorphism
φ : P → Q by x ↦→ (xφ).
Note that φ is well defined since R is invariant under φ. Now let F be the category
whose objects are the subgroups of S and for all P, Q ≤ S,
Mor F (P , Q) = {φ | φ ∈ MorF(P R, QR)}.
Then F is again a fusion system. We denote it by F/R.
Lemma 5.4.2 (Puig). Let R ✂ F. Then F/R is saturated.
Proof. This follows from [2, 8.5].
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Lemma 5.4.3. Let U ≤ R ≤ S, N = NF(U), R ✂ N and R ≤ Q ≤ NS(U). Then
AutN /R(Q/R) ∼ = NA(Q)(U)/CN A(Q)(U)(Q/R).
Proof. By 5.3.3, AutN (Q) = NA(Q)(U). Using the same notation as in 5.4.1 with N
instead of F we have that the mapping
AutN (Q) → Aut N (Q), φ ↦→ φ
is an epimorphism of groups with kernel CAutN (Q)(Q/R).
5.5 Models
Definition 5.5.1. We say F is constrained if there exists a normal centric subgroup in
F.
Example 5.5.2. Let G be a finite group, p a prime, S ∈ Sylp(G) and F = FS(G). If
G has characteristic p, then F is constrained.
Proof. Let Q = Op(G). Then by 5.3.5, Q ✂ F. In particular Q F = {Q}. Now
CS(Q) ≤ CG(Q) ≤ Q implies that Q is centric. Hence F is constrained.
Definition 5.5.3. A finite group G of characteristic p is called a model for F, if S ∈
Sylp(G) and F = FS(G).
In fact even the converse of 5.5.2 is true.
Theorem 5.5.4 (Broto, Castellana, Grodal, Levi, Oliver). Let F be constrained.
• There exists a model for F.
• If G1, G2 are models for F then there exists an isomorphism φ : G1 → G2 which
is the identity on S.
Proof. This is Proposition C in [6].
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Lemma 5.5.5. Let G be a model for F and let P ≤ S. Then P is normal in F if and
only if P is normal in G.
Proof. Assume P ✂ F and set Q = Op(G). Let g ∈ G. Then cg |Q,Q ∈ MorF(Q, Q).
Hence, since P is normal in F and F = FS(G), there is h ∈ G such that P ch = P and
ch|Q,Q = cg |Q,Q . Then h ∈ NG(P ) and gh −1 ∈ CG(Q) ≤ Q ≤ S. Since P is normal
in S, it follows g = (gh −1 )h ∈ NG(P ). Hence P ✂ G. The other direction follows
from 5.3.5.
5.6 Alperin–Goldschmidt’s Fusion Theorem
In this section let C be a set of subgroups of S.
Definition 5.6.1. • Let P, Q ≤ S and φ ∈ MorF(P, Q). We say φ is a C-
morphism if there exist sequences of subgroups of S
P = P0, P1, . . . , Pn = P φ in F, and Q1, . . . , Qn in C
and elements αi ∈ A(Qi) for i = 1, . . . , n such that Pi−1, Pi ≤ Qi, Pi−1αi = Pi
and
φ = α1|P0,P1α2|P1,P2 . . . αn|Pn−1,Pn .
• C is called conjugation family (for F) if every morphism in F is a C-morphism.
• Let Q ∈ F. Set
CQ = {P ∈ F : |P | > |Q|}
and write RF(Q) for the set of all CQ-morphism in A(Q).
Our definition of a conjugation family follows Fyn-Sydney [11]. It was suggested to
Fyn-Sydney by Shpectorov and is analogous to a definition given by Glauberman [12]
for the group case.
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Note that for Q ∈ F, every element of AutS(Q) extends to a member of A(S) and
hence AutS(Q) ≤ RF(Q). Directly from the above definition we get
Remark 5.6.2. The composition of C-morphisms is again a C-morphism.
It can be shown that F (i.e. the set of all subgroups of S) is a conjugation family for
F. So in fact, every morphism in the category F is also an F-morphism in the above
sense. Thus, this definition is consistent with our original notion of an F-morphism.
Moreover, this means that conjugation families exist. Alperin–Goldschmidt’s Fusion
Theorem will give us an even stronger result.
Definition 5.6.3. We call Q ∈ F essential if Q is centric and A(Q)/Inn(Q) has a
strongly p-embedded subgroup.
Remark 5.6.4. If Q is essential in F then every element in Q F is essential.
Proof. Since Q is centric, every element of Q F is centric. Moreover for any P ∈
Q F and φ ∈ MorF(Q, P ), the map φ ∗ : A(Q) → A(P ) as defined in 5.1.9 is an
isomorphism of groups and Inn(Q)φ ∗ = Inn(P ). Thus A(P )/Inn(P ) has a strongly
p-embedded subgroup.
This leads to the following definition:
Definition 5.6.5. An essential class is an F-conjugacy class of essential subgroups.
Theorem 5.6.6 (Alperin–Goldschmidt’s Fusion Theorem). Assume S ∈ C and C in-
tersects non-trivially with every essential class. Then C is a conjugation family.
Proof. This is a direct consequence of [16, 5.2] and [10, 2.10].
The following two results we learned from Shpectorov.
Lemma 5.6.7. Let Q ∈ F\{S}. Then the following conditions are equivalent:
(a) RF(Q) �= A(Q).
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(b) RF(Q)/Inn(Q) is a strongly p-embedded subgroup of A(Q)/Inn(Q).
(c) RF(Q)/Inn(Q) is a minimal strongly p-embedded subgroup of A(Q)/Inn(Q).
(d) Q is essential.
Proof. If P ∈ Q F is fully normalized then by 5.2.4 there exists a morphism φ ∈
MorF(NS(Q), NS(P )) such that Qφ = P . Since Q < NS(Q) it follows from 5.6.6
that φ|Q,P and φQ,P −1 are CQ-morphisms. Moreover CQ = CP since |P | = |Q|. Hence,
by 5.6.2, (φ|Q,P ) ∗ : A(Q) → A(P ) is a group homomorphism mapping RF(Q) to
RF(P ) and Inn(Q) to Inn(P ). Thus, by 5.6.4, we may from now on assume that Q
is fully normalized.
By 5.2.3, NA(Q)(RQ) ≤ RF(Q) for every QCS(Q) < R ≤ NS(Q). Thus by 2.7.5,
the properties (a) and (b) are equivalent.
Assume (a) holds. Note that Q is fully centralized since Q is fully normalized.
Hence, by 5.2.1, CS(Q) ≤ Q. Now Q is centric by 5.1.8. Therefore the two equivalent
conditions (a) and (b) imply (d). Obviously (c) implies (b). Thus it is sufficient to show
that (d) implies (c).
So assume now that Q is essential. Let R ≤ A(Q) such that AutS(Q) ≤ R and
R/Inn(Q) is a minimal strongly p-embedded subgroup of A(Q)/Inn(Q). Note that
2.7.5 gives us the existence of R and also, together with 5.2.3, that R ≤ RF(Q).
Let φ ∈ RF(Q). Then there are sequences of subgroups
Q = P0, P1, . . . , Pn = Q ∈ F and Q1, . . . , Qn ∈ CQ
and αi ∈ A(Qi) for 1 ≤ i ≤ n such that for all i = 1, . . . , n, Pi−1, Pi ≤ Qi, Pi−1αi =
Pi and φ = α1|P0,P1 . . . αn|Pn−1,Pn . We then have group isomorphisms
α ∗∗
i := (αi|Pi−1,Pi )∗ : A(Pi−1) → A(Pi) for i = 1, . . . , n.
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We set
R0 = R, Ri = Ri−1α ∗∗
i .
Then Ri/Inn(Pi) is a strongly p-embedded subgroup of A(Pi)/Inn(Pi) for every
i ∈ {0, . . . , n}. We will show by induction,
(∗) AutS(Pi) ≤ Ri for all 0 ≤ i ≤ n.
This is true for i = 0 by the choice of R. Assume now there is i < n such that
AutS(Pi) ≤ Ri. Then in particular, AutQi+1 (Pi) ≤ Ri. Using 5.1.10, we obtain
AutQi+1 (Pi+1)
∗∗
= AutQi+1 (Pi)αi+1 ≤ Riα∗∗ i+1 = Ri+1. Since Pi+1 < Qi+1, we have
Pi+1 < NQi+1 (Pi+1). Moreover, CS(Pi+1) ≤ Pi+1, since Pi+1 ∈ Q F and Q is centric.
Therefore Inn(Pi+1) < AutQi+1 (Pi+1). Hence
AutS(Pi+1)/Inn(Pi+1) ∩ Ri+1/Inn(Pi+1) �= 1.
Since Ri+1/Inn(Pi+1) is a strongly p-embedded subgroup of A(Pi+1)/Inn(Pi+1), this
yields AutS(Pi+1) ≤ Ri+1. Thus (*) holds. In particular AutS(Q) = AutS(Pn) ≤
Rn∩R = Rφ ∗ ∩R. Since R/Inn(Q) is assumed to be a strongly p-embedded subgroup
of A(Q)/Inn(Q), this implies φ ∈ R. Thus, we have shown that RF(Q) ≤ R and
therefore R = RF(Q). This proves the assertion.
Theorem 5.6.8 (Alperin–Goldschmidt’s Fusion Theorem, strong version). C is a con-
jugation family if and only if S ∈ C and C intersects non-trivially with every essential
class.
Proof. Assume C is a conjugation family and let Q be essential in F. Then by 5.6.7
there is φ ∈ A(Q)\RF(Q). Since C is a conjugation family, there exist P0, P1, . . . Pn ∈
F, Q1, . . . , Qn ∈ C and φi ∈ A(Qi) such that Pi−1, Pi ≤ Qi, Pi−1φi = Pi for i =
1, . . . n, P0 = Q = Pn and φ = φ1|P0,P1 . . . φn|Pn−1,Pn . Because of φ �∈ RF(Q) there
is i ∈ {1, . . . , n} such that |Qi| = |Q|. Then Qφ1 . . . φi = Qi and Qi ∈ Q F . Hence
Q F ∩ C �= ∅. This proves one direction and the other one follows from 5.6.6.
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Recall from the introduction that the essential rank of F is the number of conjugacy
classes of essential subgroups. So by the last result, the essential rank is crucial for the
size of a minimal conjugation family.
We conclude this section with a few consequences of Alperin–Goldschmidt’s Fu-
sion Theorem.
Lemma 5.6.9. Let A ≤ S be such that NS(A) is not fused into any essential subgroup
of F. Then A is fully normalized.
Proof. Let B ∈ A F be fully normalized. Then by 5.2.4, there exists a morphism
φ ∈ MorF(NS(A), NS(B)) such that Aφ = B. Since we assume that NS(A) is
not fused into any essential subgroup, it follows from 5.6.6 that φ extends to some
ψ ∈ A(S). Then NS(A)ψ = NS(B) and therefore |NS(A)| = |NS(B)|. So A is fully
normalized because B is fully normalized.
Lemma 5.6.10. Let P ≤ S be such that P is not fused into any essential subgroup of
F. Then every element of A(P ) extends to an element of A(S) and A(P ) is p-closed.
Proof. By 5.6.6, every element of A(P ) extends to an element of A(S) and therefore
also to an element of A(NS(P )). Now by 5.1.10, AutS(P ) is normal in A(P ). By
5.6.9, P is fully normalized. Since F is saturated, this yields AutS(P ) ∈ Sylp(A(P ))
and A(P ) is p-closed.
Lemma 5.6.11. Let P ∈ F and R ≤ NS(P ) such that RP is not fused into any
essential subgroup of F. Then every element of NA(P )(RP ) extends to an element of
A(S), and NA(P )(RP ) is p-closed.
Proof. By 5.6.9, P is fully normalized. So for every element φ of NA(P )(RP ), φ
extends to an element of MorF(Nφ, S), and RP ≤ Nφ. Then by 5.6.6, φ extends to an
element of A(S) and therefore also to an element of A(NS(P )). Hence by 5.1.10, φ
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normalizes AutS(P ) which is a Sylow p-subgroup of A(P ). This shows the assertion.
Lemma 5.6.12. The following conditions are equivalent.
(a) F has essential rank 0.
(b) S is normal in F.
(c) For each Q ≤ S, A(Q) is p-closed.
Proof. Properties (a) and (b) are equivalent by 5.6.6. By 5.6.10, (a) implies (c). If
(c) holds, then A(Q)/Inn(Q) is p-closed and cannot have a strongly p-embedded sub-
group for any Q ∈ F. So (c) implies (a).
Lemma 5.6.13. Let N ∈ F, and let D be a set of representatives of the essential
classes of F. Set C = {S} ∪ D. Then N is normal in F if and only if, for every P ∈ C,
N ≤ P and N is A(P )-invariant.
Proof. Note that by 5.6.6, C is a conjugation family. Therefore, if N ≤ P and N is
A(P )-invariant for every P ∈ C, then N is normal in F.
Suppose now N ✂ F and let P ∈ C. Then every element of A(P ) extends to an
element of A(P N). Hence, if P is essential, then N ≤ P by 5.6.7. Thus, N ≤ P for
any choice of P ∈ C. Since N is normal, N is A(P )-invariant.
Lemma 5.6.14. Suppose Q is essential in F and every essential subgroup of F is fused
into Q. Let R ∈ F such that MorF(R, Q) �= ∅. Then there is α ∈ A(S) such that
Rα ≤ Q.
Proof. Let D be the set of all essential subgroups of F contained in Q. Then by 5.6.6
and our assumption, C = {S} ∪ D is a conjugation family. In particular, if we pick an
element of MorF(R, Q), this is a C-morphism. This yields the assertion.
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5.7 Minimal fusion systems
Recall from the introduction that we call F minimal if Op(F) = 1 and for every non-
trivial fully normalized subgroup U of S the following holds:
(*) Either NF(U) has essential rank 0 or Op(NF(U)/Op(NF(U))) �= 1.
Furthermore we call F strongly minimal if Op(F) = 1 and for every non-trivial fully
normalized subgroup U of S,
(**) NF(U)/Op(NF(U)) has essential rank 0.
Remark 5.7.1. If F is strongly minimal then F is minimal.
Proof. This follows from 5.6.12.
Lemma 5.7.2. Assume Op(F) = 1 and let D be a set of representatives of the essential
classes of F. If the property (*) (respectively, (**)) holds for every non-trivial fully
normalized subgroup U of F which is contained in some element of D, then F is
minimal (respectively, strongly minimal).
Proof. Let 1 �= U ∈ F be fully normalized. If U is not fused into any element of D,
then by 5.6.6, every morphism in NF(U) extends to an element of A(S) and thus also
to an element of AutNF (U)(NS(U)). So in this case, NF(U) has essential rank 0. In
particular (*) and (**) hold.
Hence we may assume that U is fused into an element of D, i.e. there is P ∈ D
such that there exists φ ∈ MorF(U, P ). Then by 5.6.6, φ can be written as the product
of restricted automorphisms of elements of D ∪ {S}. So in particular, there exists
Q ∈ D and α ∈ A(S) such that Uα ≤ Q. Then NS(Uα) = NS(U)α, and Uα is
fully normalized since U is fully normalized. Thus, we may assume that property (*)
(respectively, (**)) holds for Uα instead of U. Also note that α|NS(U),NS(Uα) is an
isomorphism from NF(U) to NF(Uα). Hence, property (*) (respectively, (**)) holds
and F is minimal (respectively, strongly minimal).
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Lemma 5.7.3. Let F be of essential rank 1 and let Q ∈ F be essential. Then (**) holds
for Q in place of U. In particular, if Op(F) = 1 and property (*) (respectively, (**))
holds for every fully normalized subgroup 1 �= U < Q, then F is minimal (respectively,
strongly minimal).
Proof. By 5.7.2 and 5.6.12, it is sufficient to show that N = N /Op(N ) has essential
rank 0 for N = NF(Q). Note that Q is essential in N and thus by 5.6.13, Op(N ) = Q.
Set NS(Q) = NS(Q)/Q. Let 1 �= P ∈ N and assume P is the full preimage of P
in NS(Q). Then Q < P . Hence every element of AutN (P ) extends to an element of
A(S) and thus to an element of AutN (NS(Q)). By the definition of a factor system
now every element of Aut N (Q) extends to an element of Aut N (NS(Q)). Hence it
follows from 5.6.12 that N has essential rank 0.
Lemma 5.7.4. Let F be strongly minimal, Q ∈ F and U ✂ Q such that U is fully
normalized and Op(NF(U)) ≤ Q. Then [Q, Op (Op′ (A))] ≤ Op(NF(U)) for each
subgroup A of NA(Q)(U).
Proof. Set N = NF(U) and R := Op(N ). Then by assumption R ≤ Q. Moreover,
N /R has essential rank 0 because F is strongly minimal. So it follows from 5.6.12
that AutN /R(Q/R) is p-closed. Note that by 5.4.3,
AutN /R(Q/R) ∼ = NA(Q)(U)/C(Q/R).
Thus, for A ≤ NA(Q)(U), A/CA(Q/R) is isomorphic to a subgroup of AutN /R(Q/R)
and therefore p-closed. This shows Op (Op′ (A)) ≤ CA(Q/R).
Lemma 5.7.5. Let Q ∈ F be essential in F of maximal order. Let 1 �= U ≤ Q
such that U ✂ NS(Q). Set N := NF(U), U0 = Op(N ), NS(U) = NS(U)/U0 and
N = N /U0.
Then either U0 ≤ Q and Q is essential in N , or Aut N (Q) is p-closed.
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Proof. Assume Aut N (Q) is not p-closed. Then by 5.6.10, Q is N -conjugate into some
essential subgroup P of N where U0 ≤ P ≤ NS(U). Now by the definition of a factor
system, there is φ ∈ MorN (QU0, P ). Since P is essential,
Aut N (P )/Inn(P )
has a strongly p-embedded subgroup. By 5.4.3,
Aut N (P ) ∼ = NA(P )(U)/C(P/U0).
Hence NA(P )(U) is not p-closed. Therefore A(P ) is not p-closed and P is F-fused
into some essential subgroup Q0 of F. Since Q has maximal order, it follows |P | ≤
|Q0| ≤ |Q|. Hence, (QU0)φ = Qφ = P and U0 ≤ Q. In particular, Q is N -fused to P
and therefore essential in N .
Notation 5.7.6. Let Q be essential in F and 1 �= U ≤ Q such that U ✂ NS(Q). Then
we set
D(Q, U) = {U0 : U0 ≤ Q, U0 is invariant under NA(S)(U) and NA(Q)(U)}.
By U ∗ (Q) we denote the element of D(Q, U) which is maximal with respect to inclu-
sion.
Note that here U ∗ (Q) is well defined since U ∈ D(Q, U) and the product of two
elements of D(Q, U) is contained in D(Q, U). Moreover, if Op(NF(U)) ≤ Q, then
Op(NF(U)) ∈ D(Q, U) and therefore U ≤ Op(NF(U)) ≤ U ∗ (Q).
Lemma 5.7.7. Let F be minimal and Q be essential in F of maximal order. Let
1 �= U ≤ Q be such that U ✂ NS(Q), Op(NF(U)) ≤ Q and U = U ∗ (Q). Then
for every subgroup H of NA(Q)(U).
[Q, O p (O p′
(H))] ≤ U
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Proof. Set N = NF(U), N = N /U and NS(U) = NS(U)/U. Note that U = Op(N )
since Op(N ) ≤ Q and U = U ∗ (Q). Thus, if Aut N (Q) is p-closed, then the assertion
follows from 5.4.3.
Assume now Aut N (Q) is not p-closed. Then by 5.7.5, Q is essential in N . Let U0
be the full preimage of Op(N ) in NS(U). Since every element of NA(S)(U) restricts to
an element of AutN (NS(U)), U0 is NA(S)(U)-invariant.
Since Q is essential in N , it follows from 5.6.13 that U0 ≤ Q and U0 is Aut N (Q)-
invariant. Hence U0 ≤ Q and U0 is invariant under AutN (Q) = NA(Q)(U). Thus
U0 ∈ D(Q, U), U0 ≤ U ∗ (Q) = U and Op(N ) = 1. Therefore, since F is minimal, N
has essential rank 0. However, then 5.6.12 and 5.4.3 imply a contradiction to Aut N (Q)
not being p-closed.
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Chapter 6
Amalgams Realizing Fusion Systems
Throughout this chapter let p be a prime.
6.1 General results
Let A = (G1, G2, G12, φ1, φ2) be an amalgam. Recall that we write G1 ∗G12 G2 for the
free product of G1 and G2 with G12 amalgamated. From now on we always identify
G1, G2 and G12 with the obvious subgroups of G1 ∗G12 G2. Similarly we proceed for
any amalgam.
If F is a fusion system on a finite p-group S, then we say that F is realized by A,
if S is a Sylow p-subgroup of G1 and F = FS(G1 ∗G12 G2). Here a finite p-subgroup
S of a group G is called a Sylow p-subgroup of G, if every finite subgroup of G is
conjugate to a subgroup of S. We write S ∈ Sylp(G).
We will consider situations in which there is a Sylow p-subgroup S of G1 contain-
ing a Sylow p-subgroup of G2. Then S is also a Sylow p-subgroup of G1 ∗G12 G2 as it
will be part of the statement of Theorem 6.1.2.
Let B = (H1, H2, H12, ψ1, ψ2) be a further amalgam. If A and B are isomorphic,
then the corresponding free amalgamated products are isomorphic as groups. More
precisely, if we identify G1, G2 with subgroups of G = G1 ∗G12 G2, and H1, H2 with
subgroups of H = H1 ∗H12 H2, then there exists an isomorphism of groups α : G → H
87

such that G1α = H1 and G2α = H2.
If now S ∈ Sylp(G1), ˜ S ∈ Sylp(H1) and α : G → H is such an isomorphism,
then there exists h ∈ H1 such that (Sα) h = ˜ S. Hence, β = αch is an isomorphism
from G to H, and Sβ = ˜ S. Thus, by 5.1.5, FS(G) ∼ = F ˜ S (H). So we have shown the
following.
Remark 6.1.1. If two fusion systems are realized by isomorphic amalgams, then they
are isomorphic as fusion systems.
The following theorem often helps to show that a certain fusion system is realized by
an amalgam. It is stated in this form in [9]. A similar result was proved first in [18].
Theorem 6.1.2 (Robinson). Suppose there is S ∈ Sylp(G1) and T ∈ Sylp(G2) ∩
Sylp(G12) with T ≤ S. Set G = G1 ∗G12 G2. Then S ∈ Sylp(G) and
Proof. See 3.1 in [9].
FS(G) = 〈FS(G1), FT (G2)〉.
6.2 Constructing Amalgams
Hypothesis 6.2.1. Throughout this section we will assume that F is a saturated fusion
system on a finite p-group S. Furthermore we assume that F has essential rank 1
and choose an essential subgroup Q of S. We set T = NS(Q), A = NF(S) and
B = NF(Q).
As before we set
A(P ) := AutF(P ) for every P ∈ F.
Lemma 6.2.2. A and B are constrained saturated fusion systems.
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Proof. Note that by 5.6.9, Q is fully normalized in F. Clearly S is fully normalized.
Hence, by 5.3.2, A and B are saturated. By the definition of A and B, S is normal in
A, and Q is normal in B. In particular S is centric in A since CS(S) ≤ S, and Q is
centric in B since CNS(Q)(Q) ≤ CS(Q) ≤ Q. Hence A and B are constrained.
Lemma 6.2.3. F = 〈A, B〉.
Proof. This follows from our hypothesis and 5.6.6.
Lemma 6.2.4. NA(Q) = NB(T ) and NA(Q) is a constrained saturated fusion system
on T .
Proof. By 6.2.2, A is saturated. Since Q is centric and fully normalized in F, Q is also
centric in NA(Q) and fully normalized in A. Therefore by 5.3.2, NA(Q) is saturated.
Moreover, NA(Q) is also constrained since it contains Q as a normal centric subgroup.
Thus is remains to show that NA(Q) = NB(T ).
Note that both NA(Q) and NB(T ) are fusion systems on T . Now let A, B ≤ T and
φ ∈ MorF(A, B). Assume first that φ is a morphism in NA(Q). Then φ extends to
some β ∈ MorA(AQ, BQ) such that Qβ = Q. In particular φ is a morphism in B.
Now by definition of A, β extends to some α ∈ A(S). Then Qα = Qβ = Q, hence
T α = T . Thus α|NS(Q),T is an extension of φ which acts on Q and is therefore also a
morphism in B. This proves φ ∈ NB(T ).
Assume now that φ ∈ NB(T ). Then φ extends to some morphism ˆ φ ∈ AutB(T ).
Now by 5.6.6, ˆ φ extends to an automorphism of S. Therefore ˆ φ is a morphism in A.
Because ˆ φ is also a morphism in B, it follows that ˆ φ is a morphism in NA(Q). Hence
φ = ˆ φ|A,B is a morphism in NA(Q) as well.
Lemma 6.2.5. RF(Q) = NA(Q)(AutS(Q)).
Proof. Since Q is fully normalized, it follows from the saturation axioms that every
element of NA(Q)(AutS(Q)) extends to an element of A(T ). In particular, we have
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NA(Q)(AutS(Q)) ≤ RF(Q). On the other hand, since {S, Q} is a conjugation fam-
ily, every element of RF(Q) extends to an element of A(S) and therefore also to an
element of A(T ). Now by 5.1.10, RF(Q) ≤ NA(Q)(AutS(Q)).
Proof of Proposition 1. By 6.2.2 and 6.2.4, A, B and C := NA(Q) are constrained
saturated fusion systems. Therefore by 5.5.4 there exists a model G1 of A, a model
G2 of B, and a model G12 of C. Then by the definition of a model, S ∈ Sylp(G1),
T ∈ Sylp(G2) ∩ Sylp(G12), and G1, G2 have characteristic p. Moreover, by 5.5.5,
S ✂ G1 and Q ✂ G2. By 6.2.4, C = NA(Q) = NB(T ). Hence 5.3.4 yields T ∈
Sylp(NG1(Q)) ∩ Sylp(NG2(T )) and C = FT (NG1(Q)) = FT (NG2(T )). Since S ✂ G1,
T is normal in H1 = NG1(Q). So we have Op(H1) = Op(H2) = T . By 2.3.5,
CG1(Q) ≤ Q. Hence CH1(T ) ≤ CG1(Q) ≤ Q ≤ T and H1 has characteristic p.
Since G2 has characteristic p, CH2(T ) ≤ CG2(O2(G2)) ≤ O2(G2) ≤ T and H2 has
characteristic p. Therefore H1 and H2 are models for C. Hence by 5.5.4 there exist
isomorphisms φ1 : G12 → H1 and φ2 : G12 → H2 such that φ1 and φ2 are the identity
on T . Now (G1, G2, G12, φ1, φ2) is an amalgam such that (a) and (b) of Proposition 1
hold.
Note that G2/QCG2(Q) ∼ = A(Q)/Inn(Q). Since Q is essential, G2 contains there-
fore by 5.6.7 a subgroup T CG2(Q) ≤ R ≤ G2 such that R/Q is strongly p-embedded
in G2/Q. In particular O2(G2/QCG2(Q)) = 1, which implies O2(G2) ≤ QCG2(Q).
Hence by Dedekind’s Law O2(G2) = QCO2(G2)(Q) = Q since CS(Q) ≤ Q. Now note
that G2 has characteristic p and therefore CG2(Q) ≤ Q. Hence, (c) of Proposition 1
follows from 6.2.5.
The choice of G1 and G2 gives together with 6.2.3 that F = 〈FS(G1), FT (G2)〉.
Thus, (d) of Proposition 1 follows from 6.1.2.
90

Chapter 7
Examples for p = 2
For p = 2 we will later be able to classify minimal saturated fusion systems of essential
rank 1. As it turns out all examples come from groups. In this chapter we describe the
structure of these fusion systems and prove that they are actually minimal of essential
rank 1.
For the remainder of this section let G be a group, S ∈ Sylp(G) and F = FS(G).
7.1 G ∼ = P GL2(q) for some odd q
Let q be and odd prime power and set G = P GL2(q). We will regard P SL2(q) as a
subgroup of G.
Lemma 7.1.1. (a) The group S is dihedral, AutF(S) = Inn(S), F has essential
rank 1, and the essential subgroups of F are precisely the Klein fours groups
contained in S ∩ P SL2(q).
(b) If |S| ≥ 16 then F is strongly minimal.
Proof. By 2.6.8, S and S ∩ P SL2(q) are dihedral. In particular, AutF(S) = Inn(S).
By 2.6.10, F cannot have essential rank 0. Thus we may choose an essential subgroup
X of S. Then |X/(X ∩ P SL2(q))| ≤ 2 and, if Aut(X ∩ P SL2(q)) is a 2-group, then
AutF(X) is a 2-group. Hence X and X ∩ P SL2(q) are not cyclic, and not dihedral
of order greater than four. Hence X = X ∩ P SL2(q) ∼ = C2 × C2. Moreover, all
91

elementary abelian subgroups of S ∩ P SL2(q) of order 4 are conjugate under S. This
yields (a).
In particular, by 5.6.13, Op(F) is contained in X and invariant under AutF(X) ∼ =
S3. Thus, Op(F) = 1 or Op(F) = X. Assume now |S| ≥ 16. Then X is not normal
in S and thus Op(F) = 1. Let 1 �= U < X. Then NS(U) = X, or NS(U) = S and
U = Z(S). In both cases, AutNF (U)(Y ) is a 2-group for every Y ≤ NS(U). Hence,
by 5.6.12, NF(U) has essential rank 0. Thus, 5.7.3 implies (b).
7.2 S semidihedral, G ∼ = P SL2(q 2 ) : 2 for some odd q
Let q be and odd prime power and let G be the unique extension of P SL2(q 2 ) of degree
2 with semidihedral Sylow 2-subgroups.
Lemma 7.2.1. (a) We have AutF(S) = Inn(S), F has essential rank 1, and the
essential subgroups of F are precisely the Klein fours groups contained in S ∩
P SL2(q 2 ).
(b) The fusion system F is strongly minimal.
Proof. The arguments are basically the same as in the proof of Lemma 7.1.1.
7.3 G ∼ = Aut(P SL2(9)) ∼ = Aut(A6)
It is well known that Aut(P SL2(9)) ∼ = Aut(A6). Since we have formally introduced
the results about linear groups, we choose to set G = Aut(P SL2(9)). We will always
regard P GL2(9) and P SL2(9) as subgroups of G.
Lemma 7.3.1. (a) The group S is the semidirect product of a Klein fours group A
and a cyclic group C of order 8, where the action of A on C is faithful.
(b) The essential subgroups of F are precisely the elementary abelian subgroups of
S of order 8, and F has essential rank 1.
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(c) The fusion system F is strongly minimal.
Proof. Property (a) follows from 2.6.7. Moreover, S ∩ H1 ∼ = D16 and S ∩ H2 ∼ = D8
for H1 = P GL2(9) and H2 = P SL2(9).
Let Q ∈ A(S) and set V = Q ∩ H2. It is easy to check from the above facts that Q
has order 8, V has order 4, |A(S)| = 2, and S acts transitively on A(S). Moreover, it
follows that AutF(S) is a 2-group and therefore AutF(S) = Inn(S). We show next:
(7.1)
Let X ≤ S such that AutF(X) is not a 2-group. Then X is conjugate
to V or Q inside S.
Note that |X/(X ∩ H1)| ≤ 2 and |(X ∩ H1)/(X ∩ H2)| ≤ 2. Therefore it follows
from the results about coprime action that AutF(X ∩ H2) is not a 2-group. Hence,
X ∩H2 ∼ = C2 ×C2. Moreover, there is d ∈ AutF(X) of order 3 which acts irreducibly
on X ∩ H2. If X ≤ H2, then X is conjugate in S to V . Thus, we may assume that
X �≤ H2. Note that X �= S since AutF(S) is a 2-group. Hence |X| = 8. Since
CX∩H2(X) �= 1 and d acts irreducibly on X ∩ H2, [X ∩ H2, X] = 1 and X is abelian.
Therefore X = (X ∩H2)×CX(d) and X is elementary abelian. Hence X is conjugate
in S to Q. This proves (7.1).
By 2.6.10, F cannot have essential rank 0. Furthermore, V is not centric, since Q ≤
CS(V ). Hence V is not essential. Now (7.1) implies (b) and AutF(Q) ∼ = AutF(V ) ∼ =
S3. Now in particular, by 5.6.13, Op(F) ≤ Q and Op(F) is invariant under AutF(Q).
Moreover, Op(F) is normal in S, whereas V and Q are not. Therefore
(7.2)
Op(F) = 1.
Now let 1 �= U < Q. Assume NF(U) does not have essential rank 0. Let x ∈ S
such that Q �= Q x . Note that Q ≤ NS(U). In particular, V is not centric and thus not
essential in NF(U). Hence, by (7.1), Q and Q x are the only candidates for essential
93

subgroups of NF(Q).
Assume first that both Q and Q x are essential subgroups of NF(U). Then by 5.6.13,
U ≤ Q x and U is normalized by NG(Q) and NG(Q x ). This yields V = U = V x or
CQ(NG(Q)) = U = CQ x(NG(Q x )) = U x . In both cases we get a contradiction to the
choice of x and the structure of S. Thus, there is Q0 ∈ {Q, Q x } such that Q0 is the
only essential subgroup of NF(U). Then Q0 is normal in NF(U). Moreover, again
by 5.6.13, U is invariant under AutF(Q0) and therefore normal in NF(Q0). Hence,
NF(U) = NF(Q0). Now (c) follows from (7.2) and 5.7.3.
7.4 P SL3(4) with the contragredient automorphism
Set H := P SL3(4) := SL3(4) := SL3(4)/Z(SL3(4)). Then the map
σ : H → H, defined by A ↦→ (A −1 ) t for all A ∈ SL3(4),
is an automorphism of H of order 2 called the contragredient automorphism. Set
and
⎧⎛
⎪⎨
T := ⎝
⎪⎩
Note that T ∈ Syl2(H). Set
and define
G = 〈σ〉 ⋉ H
1 0 0
a 1 0
b c 1
⎛
M := ⎝
⎞
⎫
⎪⎬
⎠ : a, b, c ∈ GF (4)
⎪⎭ .
0 0 1
0 1 0
1 0 0
⎞
⎠.
τ : H → H by X ↦→ (X σ ) M for all X ∈ H.
Note that M has order 2 and (A σ ) M = (A M ) σ for all A ∈ H. Thus τ has order 2.
Moreover T M = T σ , i.e. T τ = T . Therefore
S := 〈τ〉T ∈ Syl2(G).
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Set
Lemma 7.4.1. (a) [D, τ] = 1.
⎧⎛
⎞
⎫
⎪⎨ 1 0 0
⎪⎬
V := ⎝ 0 1 0 ⎠ : a, b ∈ GF (4)
⎪⎩
⎪⎭ a b 1
,
⎧⎛
⎪⎨
E := ⎝
⎪⎩
A
⎞
⎫
0
⎪⎬
0 ⎠ : A ∈ SL2(4)
⎪⎭ 0 0 1
,
⎧⎛
⎪⎨ δ 0 0
D := ⎝ 0 δ
⎪⎩
−1 ⎞
0 ⎠ : δ ∈ GF (4)
0 0 1
∗
⎫
⎪⎬
⎪⎭ .
(b) NH(W ) ≤ NH(V ) for all 1 �= W ≤ V .
(c) NH(V ) = E ⋉ V , E ∼ = SL2(4) and V is a natural SL2(4)-module for E.
(d) NH(T ) = NH(S) = T D and NG(T ) = SD. In particular, NG(T ) is p-closed.
(e) T = V V τ , V ∩ V τ = Z(T ) and V # ∪ (V τ ) # is the set of involutions in T .
(f) Z(T ) = Z(S).
Proof. An elementary calculation shows (a). Parts (b) and (c) follow from 2.5.4. In
particular, (b) implies together with Z(T ) ≤ V that NH(T ) ≤ NH(Z(T )) ≤ NH(V ).
Now by (c), NH(T ) = NE⋉V (T ) = T D. Since 〈τ〉 ≤ NG(T ) ≤ G = 〈τ〉H, it follows
that NG(T ) = 〈τ〉NH(T ) = SD. Thus (d) holds.
Claim (e) follows from (c), 3.4.5(b) and 3.4.6. By (a), D normalizes [Z(T ), τ]
which is a proper subgroup of Z(T ). Since (c) implies that D acts irreducibly on
Z(T ), it follows [Z(T ), τ] = 1. This shows (f).
Lemma 7.4.2. Let 1 �= X ≤ T .
(a) If Ω(Z(X)) ≤ V then NH(X) ≤ NH(V ).
(b) If X �≤ V and X �≤ V τ then NH(X) ≤ NH(T ).
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(c) Let 1 �= W < V . Then NG(W ) is p-closed.
Proof. If Ω(Z(X)) ≤ V , then 7.4.1(b) implies that NH(X) ≤ NH(Ω(Z(X))) ≤
NH(V ). Thus, (a) holds.
For the proof of (b) assume X �≤ V and X �≤ V τ . By 7.4.1(e), Ω(Z(X)) ≤ V or
Ω(Z(X)) ≤ V τ . Since T τ = T , we can replace X by X τ if necessary and assume
Ω(Z(X)) ≤ V . Then by (a), NH(X) ≤ NH(V ). Now the structure of NH(V ) implies
that NH(X) ≤ NH(T ). Thus, (b) holds.
By (a) and 7.4.1(c), there exists T1 ∈ Sylp(H) such that T1 ≤ NH(W ) ≤ NH(T1).
Then by the Frattini Argument, T1 is normal in NG(W ). Thus, (c) follows from
7.4.1(d).
Lemma 7.4.3. (a) F has essential rank 1 and V is essential in F.
(b) F is strongly minimal.
Proof. Set F = FS(G). By 7.4.1(c), V is essential in F. Let Y ≤ S and assume Y is
essential in F. Then
(*) CS(Y ) ≤ Y , and
(**) NG(Y )/Y CG(Y ) has a strongly 2-embedded subgroup.
Set X := Y ∩T . If X = 1 then |Y | = 2 and AutF(Y ) = Aut(Y ) = 1, a contradiction.
Thus X �= 1. Moreover Y �= S and by 7.4.1(d), Y �= T . Thus we have X �= T .
Assume first X �≤ V and X �≤ V τ . Then by 7.4.2(b), NH(X) is 2-closed and
T1 := NT (X) ∈ Syl2(NH(X)). Therefore, by the Frattini Argument, T1 is normal
in NG(Y ). Hence by (**), T1 ≤ Y CG(Y ). Now (*) yields T1 ≤ Y CG(Y ) ∩ S =
Y CS(Y ) = Y , i.e. NT (X) ≤ X. However, this implies X = T , a contradiction as
above.
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Thus X ≤ V or X ≤ V τ . Replacing Y by Y τ if necessary we may assume that
X ≤ V . If X < V then 7.4.2(c) yields a contradiction to (**). Hence X = V and
therefore Y = V as required. This shows (a).
Observe that every non-trivial normal subgroup of S contained in V is not invariant
under AutF(V ). Thus by 5.6.13, Op(F) = 1. Furthermore, by 7.4.2(c), NG(U) is p-
closed for every 1 �= U < V . Hence (b) follows from (a), 5.3.4 and 5.7.3.
97

Chapter 8
Recognizing SL2(q) in Fusion Systems
8.1 Notation and main results of this chapter
In this section let F be a saturated fusion system on a finite p-group S. For every
P ≤ S, set
Let Q ∈ F be essential. Set
A(P ) = AutF(P ),
A ◦ (P ) = O p′
(A(P )).
Z = Ω(Z(S)) and T = NS(Q).
Note that Z ≤ Ω(Z(Q)) since Q is centric. Let V be an A ◦ (Q)-invariant subgroup of
Ω(Z(Q)) containing Z.
The main theorem we are going to prove in this section is the following:
Theorem 8.1.1. Suppose every essential subgroup of F is fused into Q. Furthermore
assume J(S) �≤ Q and Z is not A(Q)-invariant. Then the following hold:
(a) CS(V ) = Q, NS(J(Q)) = T and J(T ) �≤ Q.
(b) CA(Q)(V )/Inn(Q) is a p ′ -group.
(c) A ◦ (Q)/CA ◦ (Q)(V ) ∼ = SL2(q).
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(d) V/CV (A ◦ (Q)) is a natural SL2(q)-module for A ◦ (Q)/CA ◦ (Q)(V ).
Applying 3.2.3, 3.2.7 and 3.4.3 we obtain the following corollary of 8.1.1:
Corollary 8.1.2. Assume the hypothesis of 8.1.1. Then the following properties hold:
(a) |V/CV (A)| = |A/CA(V )| = q, for every offender A ≤ T on V . In particular,
this holds for every A ∈ A(T ) with A �≤ Q.
(b) For all A ∈ A(T ), CA(V )V = (A ∩ Q)V ∈ A(Q).
(c) A(Q) ⊆ A(T ).
Note that, by 5.6.13, Theorem 8.1.1 applies in particular if F has essential rank 1 and Z
and J(S) are not normal in F. Moreover, if F has essential rank 1 and G2 is a model
for NF(Q), then the map G2 → A(Q) defined by g ↦→ cg |Q,Q is an epimorphism.
Therefore, Theorem 1 (see Introduction) is a consequence of Theorem 8.1.1.
In order to prove Theorem 8.1.1 we introduce one further definition. We say that
U is F-characteristic in Q and write
U charF Q
if U ≤ Q, U ✂ T and A ◦ (Q) = CA ◦ (Q)(Ω(Z(Q)))NA ◦ (Q)(U).
We remind the reader that in 5.2.2 we defined for U ∈ F,
RU = {cg |U,U : g ∈ R} for each R ≤ NS(U).
8.2 Preliminary results
Assume from now on that Q is fully normalized. Recall that we defined in 5.6.1 the
subgroup RF(Q) of A(Q).
Remark 8.2.1. Assume RF(Q)CA(Q)(V ) �= A(Q). Then the following hold:
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(a) CA(Q)(V )/Inn(Q) is a p ′ -group and RF(Q)CA(Q)(V )/CA(Q)(V ) is a strongly
p-embedded subgroup of A(Q)/CA(Q)(V ).
(b) CS(V ) = Q.
Proof. Part (a) follows directly from 5.6.7 and 2.7.10. In particular, we have that
CT (V )Q = Inn(Q) and Q = CT (V ). Since CS(V ) is nilpotent, this implies (b).
Lemma 8.2.2. Assume J(T ) �≤ Q and RF(Q)CA(Q)(V ) �= A(Q). Then the following
hold:
(a) CA(Q)(V )/Inn(Q) is a p ′ -group and CS(V ) = Q.
(b) A ◦ (Q)/CA ◦ (Q)(V ) ∼ = SL2(q) for q = |T/Q|.
(c) V/CV (A ◦ (Q)) is a natural SL2(q)-module for A ◦ (Q)/CA ◦ (Q)(V ).
Proof. Set A(Q) = A(Q)/CA(Q)(V ). Note that (a) is a consequence of 8.2.1. Since
J(T ) �≤ Q, it follows from 3.2.13 that O A(Q) (V ) �= ∅. Hence, also O A ◦ (Q) (V ) �= ∅.
On the other hand, again by 8.2.1, A(Q) has a strongly p-embedded subgroup.
Hence, by 2.7.9, A ◦ (Q) has a strongly p-embedded subgroup. Now by 3.5.8, A ◦ (Q) ∼ =
SL2(q) for q = |AutS(Q)| = |T/Q|, and V/CV (A ◦ (Q)) is a natural SL2(q)-module
for A ◦ (Q).
Remark 8.2.3. Let P, R ≤ S such that J(P ) ≤ R and J(R)α ≤ P for some α ∈
MorF(J(R), S). Then J(R) = J(P ) = J(R)α = J(P )α.
Proof. Let A ∈ A(P ) and B ∈ A(R). Then
|A| ≤ |B| = |Bα| ≤ |A|,
and so |A| = |B|. Thus J(P ) ≤ J(R) and J(R)α ≤ J(P ). This yields J(R) =
J(P ) = J(R)α = J(P )α.
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In view of 8.2.2, we are in particular interested in showing J(T ) �≤ Q. We analyze this
problem under the following hypothesis which we assume for the remainder of this
section.
Hypothesis 8.2.4. (a) Z is not F-characteristic in Q, and
(b) J(S) �≤ Q.
Remark 8.2.5. J(NS(J(Q))) �≤ Q.
Proof. Assume J(NS(J(Q))) ≤ Q. Then J(Q) = J(NS(J(Q))) is normalized by
NS(NS(J(Q))) and hence, NS(J(Q)) = S. Therefore, J(S) ≤ Q, a contradiction to
Hypothesis 8.2.4(b).
Thus, in order to show J(T ) �≤ Q, it is sufficient to show that NS(J(Q)) = T . To
approach this problem, we look at the set C0(Q) consisting of all those subgroups U of
Q with the following properties:
• U charF Q,
• Z ≤ U, and
• J(NS(U)) is not F-conjugate into any essential subgroup in F.
Remark 8.2.6. NA(U)(Z) < A(U) for U ∈ C0(Q).
Proof. Note that Z ≤ U since U ∈ C0(U). Moreover, every element of NA ◦ (Q)(U)
acts on U. Hence, if Z were normalized by A(U), then Zφ = Zφ|U,U = Z for
every φ ∈ NA ◦ (Q)(U). Therefore, NA ◦ (Q)(U) ≤ NA ◦ (Q)(Z), and Z would be F-
characteristic in Q. This is a contradiction to Hypothesis 8.2.4(a).
By 8.2.6, there exists a subgroup P of A(U) such that AutS(U) ≤ P and P �≤
NA(U)(Z). Such a subgroup P , which is minimal with these properties, will play a
crucial role in the next lemma, the main result of this section.
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Lemma 8.2.7. 1 Let U ∈ C0(Q), set T1 = NS(U), T0 = (T1)U and A = A(U). Assume
J(T1) �≤ Q.
Let P be a subgroup of A which is minimal with the property that T0 ≤ P and
P �≤ NA(Z). Set W = 〈Z P 〉 and P = P/CP (W ). Then the following conditions
hold:
(a) CT1(W )U = Op(P ) ∈ Sylp(CP (W )) and J(T1) �≤ CS(W ).
(b) ∅ �= {BU : B ∈ A(T1) and [W, B] �= 1} ⊆ OP (W ).
(c) A P (W ) is the set of minimal by inclusion elements of O P (W ).
(d) There is B ∈ A(T1) such that BU ∈ A P (W ).
Moreover, one of the following holds:
(I) P ∼ = SL2(q), W/CW (P ) is a natural SL2(q)-module for P , and CT1(W ) is
fused into some essential subgroup in F.
(II) For each B ∈ A(T1) with BU ∈ A P (W ), BCT1(W ) is fused into some essential
subgroup in F.
Moreover, there are subgroups E1, . . . , Er of P such that the following hold for
all i = 1, . . . , r:
– P = (E1 × . . . × Er)T0,
– T0 acts transitively on {E1, . . . , Er},
– W = CW (E1 × . . . × Er) �
i [W, Ei],
– Ei ∼ = SL2(p n ) for some n ∈ N, or p = 2 and Ei ∼ = S5,
– [W, Ei]/C[W,Ei](Ei) is a natural module for Ei.
1 Recall the notation introduced in 3.2.8 and 3.5.1.
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Proof. Since U ∈ C0(Q), J(T1) is not fused into any essential subgroup. Hence,
every subgroup R of T1 with J(T1) ≤ R is not fused into any essential subgroup. In
particular, T1 is not fused into any essential subgroup. Thus, 5.6.9 yields:
(8.1)
U is fully normalized.
Furthermore, 5.6.11 implies the following property:
(8.2)
The elements of NA(RU) extend to elements of A(S), and NA(RU)
is p-closed, for every subgroup R of T1 with J(T1) ≤ R.
It follows from (8.2) that NA(T0) ≤ NA(Z). Hence, by 2.8.3, P is minimal parabolic
and NP (Z) is the unique maximal subgroup of P containing T0.
Let Q0 ≤ T1 be maximal such that (Q0)U = Op(P ). Then U ≤ Q0. If J(T1) ≤ Q0,
then by (8.2) every element of P extends to an element of A(S). This is a contradiction
to P �≤ NA(Z). Therefore, J(T1) �≤ Q0.
Since CP (W ) is a normal subgroup of P , by 2.8.2, either O p (P ) ≤ CP (W ), or
CT0(W ) is normal in P . If O p (P ) ≤ CP (W ), then P = O p (P )T1 ≤ CP (Z) ≤ NA(Z),
a contradiction. Hence, CT0(W ) ≤ Op(P ) and CT1(W ) ≤ Q0. On the other hand, W
is centralized by Op(P ), since Z is centralized by Op(P ). This shows (a). Now 3.2.13
implies (b). This yields together with (8.2) that
(8.3)
NP (JT0(W )CP (W )) is p-closed.
Let Q1 be the full preimage of Op(P ) in P . Since P is not a p-group, O p (P ) �≤ Q1.
Hence it follows from 2.8.2 that Q1 ∩ T0 ≤ Op(P ). Now (a) implies Op(P ) = 1.
Note that P is minimal parabolic and N P (Z) is the unique maximal subgroup of
P containing T0, since NP (Z) is the unique maximal subgroup of P containing T0
and CP (W ) ≤ CP (Z). Moreover, it follows from Z ≤ CW (T0) that C P (CW (T0)) ≤
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C P (Z) ≤ N P (Z). Therefore we can apply Theorem 3.5.5. This yields that for D =
A P (W ) there exist subgroups E1, . . . , Er of P containing CP (W ), such that for 1 ≤
i ≤ r the following hold:
(i) P = (E1 × . . . × Er)T0,
(ii) T0 acts transitively on {E1, . . . , Er},
(iii) D = (D ∩ E1) ∪ . . . ∪ (D ∩ Er),
(iv) W = CW (E1E2 . . . Er) � r
i=1 [W, Ei], with [W, Ei, Ej] = 1 for j �= i,
(v) Ei ∼ = SL2(p n ), or p = 2 and Ei ∼ = S2 n +1, for some n ∈ N,
(vi) [W, Ei]/C[W,Ei](Ei) is a natural module for Ei.
This implies together with 3.2.7 and 3.4.3 that |W/CW (A)| = |A| for every A ∈ D. In
particular, by the definition of D, m P (W ) = |W |. Hence, there is no over-offender in
OP (W ) on W and (c) holds.
Note that a subgroup D of T1 is an offender on W if and only if DU is an offender
on W . Therefore, by 3.2.6, there exists B ∈ A(T1) such that BU is an offender on W
which is minimal with respect to inclusion. Now (d) follows from (c). We show next:
(v’) For every 1 ≤ i ≤ r, Ei ∼ = SL2(p n ) for some n ∈ N, or Ei ∼ = S5.
For the proof of (v’) we may assume p = 2 and Ei ∼ = S2 n +1 for each 1 ≤ i ≤ r
and some n ∈ N. By (8.3), NP (JT0 (W )) is p-closed. By 3.5.6, NEi (J (W )) ≤
Ei∩T0
N P (J T0 (W )). Hence it follows from 3.3.2 that Ei ∼ = S3 or S5. Since S3 ∼ = SL2(2),
this yields (v’).
Assume now (II) does not hold. Then by (i)-(vi) and (v’), we may assume that
BQ0 is not fused into any essential subgroup of F. Hence, by 5.6.11, NP ((BQ0)U) is
p-closed. Note that (BQ0)U ∈ Sylp(BUCP (W )). Therefore by the Frattini Argument,
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N P (BU) is p-closed as well. Observe also that, by (iii), there exists k ≤ r such that
BU ≤ Ek. Now, if r �= 1, then for j �= k, Ej is p-closed, a contradiction to (v’).
Therefore r = 1. If E1 ∼ = S5, then by 3.3.2, BU corresponds to a transposition in S5
and C Ek (BU) ∼ = C2 × S3. This is again a contradiction to N P (BU) being p-closed.
Hence, E1 ∼ = SL2(q) for some power q of p. Since Op(P ) = 1, it follows from
2.6.11 that
P ∼ = SL2(q).
In particular, P is not p-closed and therefore by 5.6.11, Q0 = CT1(W ) is fused into
some essential subgroup of F. This shows (I) and completes the proof of Lemma
8.2.7.
Lemma 8.2.8. Assume J(Q) ∈ C0(Q). Set U = J(Q) and choose the notation as in
8.2.7.
(a) Suppose (I) of Theorem 8.2.7 holds, and there exists an automorphism α ∈ A(S)
such that CT1(W )α ≤ Q. Then T1 = T .
(b) For every B ∈ A(T1) with [W, B] �= 1, BU is not fused into Q.
Proof. Set Q0 = CT1(W ). For the proof of (a) assume that (I) of Lemma 8.2.7 holds
and there is a morphism α ∈ A(S) such that Q0α ≤ Q. Note that U = J(Q) ≤ Q0.
Now 8.2.3 yields Uα = U. Therefore also T1α = T1. Since P/CP (W ) ∼ = SL2(q)
and T1/Q0 ∼ = T0/(Q0)U ∼ = T0/(T0 ∩ CP (W )), T1/Q0 is abelian. Thus, also T1/Q0α
is abelian and [T1, Q] ≤ Q0α ≤ Q. This shows (a).
Assume now there is B ∈ A(T1) with [W, B] �= 1 and MorF(BU, Q) �= ∅. Then
by 8.2.3, J(BU) = J(Q) = U. This implies B ≤ U which contradicts the assumption
[B, W ] �= 1.
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8.3 The proof of Theorem 8.1.1
From now on assume that every essential subgroup of F is fused into Q. Furthermore
suppose J(S) �≤ Q and Z is not A(Q)-invariant.
Observe that it follows from the Frattini Argument and 5.6.6, that Z is not A ◦ (Q)-
invariant. Hence, Z is not F-characteristic in Q.
8.3.1. J(Q) ∈ C0(Q).
Proof. Clearly J(Q) charF Q and Z ≤ J(Q). By 8.2.5, J(NS(J(Q))) �≤ Q. Thus,
8.2.3 implies that J(NS(J(Q))) is not fused into Q and therefore, by hypothesis, not
into any essential subgroup of F.
8.3.2. T = NS(J(Q)) and J(T ) �≤ Q.
Proof. We have shown in 8.3.1 that J(Q) ∈ C0(Q). Thus, we can apply 8.2.7 to
U = J(Q). By 8.2.8(b), (II) of 8.2.7 cannot hold. So (I) of 8.2.7 holds. Hence, by
5.6.14 and 8.2.8(a), T = NS(U). Now 8.2.5 yields J(T ) �≤ Q.
Proof of Theorem 8.1.1. By our hypothesis and 5.6.6, every element of RF(Q) extends
to an element of A(S). Hence, Z is normalized by RF(Q). Since Z is not A ◦ (Q)-
invariant, it follows that A(Q) �= RF(Q)CA(Q)(V ). Now 8.3.2 and 8.2.2 yield the
assertion.
106

Chapter 9
Pushing Up in Fusion Systems
9.1 Notation, basic properties and main results of this
chapter
Throughout this chapter, F is a saturated fusion system on a finite p-group S. Further-
more, F is always assumed to be of essential rank 1 and Op(F) = 1. Let Q be an
essential subgroup of F, T = NS(Q) and q = |T/Q|.
We will use without reference that by 5.2.4, Q is fully normalized. We continue to
use the notation and definitions introduced in Section 8.1. Moreover, we set
V (P ) = 〈Z A(P ) 〉
for every subgroup P of S containing Z. Note that V (U) ≤ V (Q) for every F-
characteristic subgroup U of Q with Z ≤ U. Recall that we defined in 5.7.6
D(P, U) = {U0 : U0 ≤ P, U0 is invariant under NA(S)(U) and NA(P )(U)}
for P ∈ Q F and every subgroup U of P with U ✂ NS(P ). Furthermore, U ∗ (P ) is the
maximal element of D(P, U).
For P ∈ Q F we set
C(P ) = {U ≤ P : U charF P, Z ≤ U, CS(V (U)) = U = U ∗ (P )}.
Let A = (G1, G2, G12, φ1, φ2) be the amalgam for F as described in Lemma 1. Set
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M := Op′ (G2). By the construction of A,
α : G2 → A(Q), defined by g ↦→ cg |Q,Q
is an epimorphism. In particular, Mα = Op′ (G2)α = A◦ (Q). We will use without
reference that A(Q) is an epimorphic image of G2 in this way.
Lemma 9.1.1. Let U ≤ Q be A ◦ (Q)-invariant and A(S)-invariant. Then U = 1.
Proof. Since Op(F) = 1, it follows from 5.6.13 that no non-trivial subgroup of Q is
invariant under A(S) and A(Q). By the Frattini Argument, A(Q) = A ◦ (Q)NA(Q)(TQ).
Moreover, by 5.6.11, every element of NA(Q)(TQ) extends to an element of A(S) and
therefore normalizes U.
Lemma 9.1.2. Z is not F-characteristic in Q and J(S) �≤ Q.
Proof. This follows from 9.1.1.
Now Theorem 1 and 8.1.2 give us for every normal subgroup V of M with Z ≤ V ≤
Ω(Z(Q)) the following properties:
(I) NS(J(Q)) = T , J(T ) �≤ Q and CS(V ) = Q,
(II) A ◦ (Q)/CA ◦ (Q)(V ) ∼ = M/CM(V ) ∼ = SL2(q),
(III) V/CV (M) is a natural SL2(q)-module for M/CM(V ),
(IV) A(Q) ⊆ A(T ) and V (A ∩ Q) ∈ A(Q) for every A ∈ A(T ).
In particular, we will use these properties for V = V (Q) and V = Ω(Z(Q)).
As the main result of this chapter, we will show Theorem 2. Crucial for that is the
following lemma.
Lemma 9.1.3. Let F be minimal and 1 �= X charF Q. Then NS(X) = T .
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Moreover, the minimality of F enables us to show an improved version of 9.1.1 (see
9.2.7). Using this, we obtain the following corollary of Lemma 9.1.3.
Corollary 9.1.4. Let F be minimal. Then M �= CM(V (Q))NM(C) for every non-
trivial characteristic subgroup C of T .
For the proof of Lemma 9.1.3 and Corollary 9.1.4 see Sections 9.3 and 9.4. Corol-
lary 9.1.4 will enable us to apply the results of Section 4.1 and prove Theorem 2.
9.2 Preliminary results
Lemma 9.2.1. Let U ≤ Q such that U ✂ T and Z ≤ U. Set A := NA(Q)(U). Then
CS(U) ≤ CS(V (U)) ≤ Q, or Z is A-invariant.
Proof. Set W = V (U). If CS(W ) �≤ Q, then CT (W ) �≤ Q and by 5.6.11, Z is
normalized by NA(Q)(CT (W )Q). The Frattini Argument yields
so A normalizes Z.
A = CA(W )NA(CT (W )Q),
Lemma 9.2.2. Let F be minimal and 1 �= U ≤ Q such that U ✂ T and Op(NF(U)) ≤
Q. Assume U = U ∗ (Q). Then the following hold:
(a) [Q, Op (Op′ (H))] ≤ U for every subgroup H of NA(Q)(U).
(b) Z ≤ U and U = Op(NF(U)).
(c) CS(U) = CS(V (U)) = U or Z is invariant under NA(Q)(U).
Proof. Property (a) follows from 5.7.7. Set H := NA(Q)(U). Then (a) implies that
[UZ, Op (Op′ (H))] ≤ U ≤ UZ. Moreover, by 5.6.11, every element of NH(TQ)
extends to an element of A(S) and therefore acts on Z. The Frattini Argument yields
H = Op′ (H)NH(TQ) = Op (Op′ (H))NH(TQ). Hence UZ is H-invariant. Clearly UZ
109

is invariant under NA(S)(U) and therefore UZ ∈ D(Q, U). Thus Z ≤ UZ ≤ U ∗ (Q) =
U. Also note that Op(NF(U)) ∈ D(Q, U) and hence Op(NF(U)) = U. This shows
(b).
By 9.2.1, Z is invariant under NA(Q)(U), or CS(U) and CS(V (U)) are elements of
D(Q, U). This implies (c).
Lemma 9.2.3. Let F be minimal and 1 �= U charF Q. Set A = NA ◦ (Q)(U). Then the
following conditions hold:
(a) Op(NF(U)) charF Q and Op(NF(U)) is A-invariant.
(b) If Z ≤ U then CS(V (U)) and CS(U) are F-characteristic in Q and A-invariant.
(c) Op(NF(U ∗ (Q))) = U ∗ (Q).
(d) [Q, Op (Op′ (H))] ≤ U ∗ (Q) for every subgroup H of NA(Q)(U).
(e) U ∗ (Q)X charF Q for every subgroup X of Q which is normal in T .
Proof. Note that A ≤ AutNF (U)(Q). Therefore every element of A extends to an
element of A(Op(NF(U))Q). Since U charF Q, A is not p-closed. Hence, by 5.6.6,
Op(NF(U)) ≤ Q. This implies (a). Now (b) is a consequence of (a), 9.1.2 and 9.2.1.
Note that U ∗ (Q) charF Q. Hence, (a) applied to U ∗ (Q) in place of U gives us
Op(NF(U ∗ (Q))) ≤ Q. Thus, Op(NF(U ∗ (Q))) ∈ D(Q, U). Now the maximality of
U ∗ (Q) yields (c). Clearly, U ∗ (U ∗ (Q)) = U ∗ (Q), so 9.2.2(a) applied to U ∗ (Q) gives
(d).
Now let X ≤ Q such that X is normal in T . Then U ∗ (Q)X is normal in T and
by (d), U ∗ (Q)X is invariant under Op′ (A) = Op (Op′ (A))TQ. Since U charF Q,
A ◦ (Q) = ACA ◦ (Q)(V (Q)) = O p′
(A)CA ◦ (Q)(V (Q)). Hence (e) holds.
Corollary 9.2.4. Let F be minimal and 1 �= U charF Q. Then U ∗ (Q) ∈ C(Q).
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Proof. We may replace U by U ∗ (Q) and assume that U = U ∗ (Q). Then by 9.2.3(c),
U = Op(NF(U)). It follows now from 9.2.2(b),(c) and 9.1.2 that Z ≤ U and U =
CS(V (U)). Hence U ∈ C(Q).
Definition 9.2.5. Let W ≤ S be elementary abelian and W ≤ Y ≤ NS(W ) such
that [W, J(Y )] �= 1. Then we write A∗(Y, W ) for the set of elements A ∈ A(Y ) with
[A, W ] �= 1 for which ACY (W ) is minimal with respect to inclusion.
Lemma 9.2.6. Let U ∈ C(Q), W := V (U) and A ∈ A∗(T, W ). Assume A �≤ Q. Then
|W/CW (A)| = |A/CA(W )| = q and W = V (Q)CW (A).
Proof. It follows from (I) that [V (Q), A] �= 1 and from (IV) that
Hence the assertion follows from 3.2.5.
|V (Q)/CV (Q)(A)| = |A/CA(V (Q))| = q.
Lemma 9.2.7. Assume F is minimal, U charF Q and U is A(S)-invariant. Then
U = 1.
Proof. Assume U �= 1. Set H1 = NA ◦ (Q)(U) and H2 = NA ◦ (Q)(Z). Since U charF Q,
it follows
A ◦ (Q) = H1CA ◦ (Q)(V (Q)) = H1H2.
Set U1 = U ∗ (Q) and U2 = Z ∗ (Q). Since Z and U are A(S)-invariant, TQ ≤ Hi for
i = 1, 2. Moreover, Op′ (Hi) = TQOp (Op′ (Hi)) and by the Frattini Argument
Hence,
Hi = O p′
(Hi)NHi (TQ) = O p (O p′
(Hi))NHi (TQ).
(∗) A ◦ (Q) = 〈O p (O p′
(H1)), O p (O p′
(H2)), NA◦ (Q)(TQ)〉.
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Set U ◦ = U1U2. Since U and Z are A(S)-invariant, it follows that U1, U2 and U ◦ are
A(S)-invariant as well. In particular, since every element of NA ◦ (Q)(TQ) extends to an
element of A(S), U ◦ is normalized by NA ◦ (Q)(TQ).
By 9.2.3(a), Op(NF(U1)) ≤ Q. If Op(NF(U2)) �≤ Q, then every element of H2
extends to an element of A(S) and therefore normalizes U. Thus, U is invariant under
A ◦ (Q) = H1H2, a contradiction to 9.1.1. Hence, Op(NF(U2)) ≤ Q. Therefore, we
can apply 9.2.2(a) to Ui and get
[U ◦ , O p (O p′
(Hi))] ≤ [Q, O p (O p′
(Hi))] ≤ Ui ≤ U ◦
for i = 1, 2. Thus, by (*), U ◦ is A ◦ (Q)-invariant. Since U ◦ is also A(S)-invariant, this
is a contradiction to 9.1.1.
Lemma 9.2.8. Let F be minimal, 1 �= U charF Q and g ∈ G1. Furthermore assume
either
(*) U g = U, or
(**) F is strongly minimal and U charF Q g .
Then [Q, Op (Op′ (H))] g ≤ Q for every subgroup H of NM(U).
Proof. Let H ≤ NM(U). Note that by 5.6.9, U is fully normalized. If (**) holds,
then it follows from 5.7.4 and 9.2.3(a) that [Q, Op (Op′ (H))] g = [Qg , Op (Op′ (Hg ))] ≤
Op(NF(U)) ≤ Q.
Thus we may assume that (**) holds. Then U = U g charF Q g . Moreover,
U ∗ (Q g ) ≤ Q g and U ∗ (Q) is invariant under NA(S)(U). Now cg ∈ NA(S)(U) and
the definition of U ∗ (Q g ) yield U ∗ (Q g ) = U ∗ (Q g ) g−1
.
≤ Q. Thus, by 9.2.3(d),
[Q, O p (O p′
(H))] g = [Q g , O p (O p′
(H g ))] ≤ U ∗ (Q g ) ≤ Q
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Lemma 9.2.9. Let F be minimal. Assume there is V (Q) ≤ T1 ≤ T and T1 ≤ H ≤ M
such that T1 ∈ Sylp(H) and M = CM(V (Q))H. Moreover, assume there exists
T ≤ T0 ≤ S such that T1 char T0 and
(∗) NS(U) = T0 for all 1 �= U ≤ T1 ∩ Q with U ✂ T0 and U charF Q.
(a) H has the Pushing Up Property and Op(H) = Q ∩ T1.
(b) Let [Q ∩ T1, O p (H)] ≤ V (Q). Then Q ∩ T1 is elementary abelian and Q ∩
T1/CQ∩T1(H) is a natural SL2(q)-module for H/CH(V (Q)).
(c) Assume again [Q ∩ T1, O p (H)] ≤ V (Q). Furthermore suppose that one of the
following holds:
(i) F is strongly minimal and T1 = T , or
(ii) p = 2 and T1 = T , or
(iii) p = 2, T1 char T �= S and J(Q) ≤ T1.
Then |CQ∩T1(H)| ≤ q and |Q ∩ T1| ≤ q 3 .
Proof. Note that Q∩H = Q∩T1 since T1 ∈ Sylp(H). Hence, Q∩T1 ✂H. Moreover,
Op(H) ≤ Q since otherwise every element of HQ would extend to an element of A(S)
and normalize Z, a contradiction to 9.1.2. Hence, Op(H) = T1 ∩ Q. Note also that
|T1/T1 ∩ Q| = q and therefore T = T1Q.
Assume now there is 1 �= U ≤ T1 ∩ Q such that U ✂ NG1(T1) and U ✂ H.
Since T1 char T0, U ✂ NG1(T0) and in particular, U ✂ T0 and U ✂ T . Therefore,
A ◦ (Q) = C(V (Q))HQ = C(V (Q))NA ◦ (Q)(U) implies U charF Q. Thus by (*),
NS(U) = T0. On the other hand, U ✂ NS(T0). Hence, T0 = NS(T0), i.e. S = T0.
Then U is A(S)-invariant and F-characteristic in Q. This is a contradiction to 9.2.7.
Thus we have shown the following
(9.1)
Let 1 �= U ≤ Q ∩ T1 such that U ✂ NG1(T1). Then U is not normal in H.
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In particular, (a) holds. To ease notation we assume from now on that H = Op′ (H).
(This is possible because otherwise we can replace H by Op′ (H).)
For the proof of (b) and (c) assume [Q ∩ T1, O p (H)] ≤ V (Q). Then we have
that [Q ∩ T1, O p (H)] = [Q ∩ T1, O p (H), O p (H)] ≤ V := [V (Q), O p (H)]. Set
X := 〈V NG 1 (T1) 〉 and note that X ≤ T1. Assume X ≤ Q. Then X ≤ Q ∩ T1
and [X, O p (H)] ≤ V ≤ X, i.e. X ✂ H = O p (H)T1. This is a contradiction to (9.1).
(9.2)
Therefore, X �≤ Q, i.e. we can choose g ∈ NG1(T1) such that
V g �≤ Q.
Assume there is 1 �= U0 ≤ Q ∩ T1 such that U0 is normal in T and H. Then
U0 charF Q. If U g
0 = U0, then 9.2.8 yields V g ≤ [Q, O p (H)] g ≤ Q, a contradic-
tion to the choice of g. Thus we have shown the following.
(9.3)
Let 1 �= U0 ≤ Q ∩ T1 such that U0 is normal in T and H. Then U g
0 �= U0.
Note that V/CV (H) is a natural SL2(q)-module for H/CH(V (Q)). Therefore, there is
no over-offender in T1 on V or V g . In particular, V g does not act as an over-offender
on V and V does not act as an over-offender on V g . Therefore,
|V/CV (V g )| = |V g /CV g(V )|.
So V g acts as an offender on V and vice versa, i.e.
(9.4)
|V/CV (V g )| = |V g /CV g(V )| = q and T1 = V (Q g ∩ T1) = V g (Q ∩ T1).
Now set Q0 = Q ∩ Q g ∩ T1. With a Dedekind argument it follows from T1 = V (Q g ∩
T1) = V g (Q ∩ T1) that Q ∩ T1 = V Q0 and Q g ∩ T1 = V g Q0. Hence, Φ(Q ∩ T1) =
Φ(Q0) = Φ(Q g ∩ T1). Now (9.3) yields Φ(Q ∩ T1) = 1. Thus,
(9.5)
Q ∩ T1 is elementary abelian.
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In particular, [Q ∩ T1, O p (H)(Q ∩ T1)] ≤ V (Q). Since CH(V (Q))/Q ∩ T1 is a p ′ -
group, we have CH(V (Q)) ≤ O p (H)(T1 ∩ Q) and then also [Q ∩ T1, CH(V (Q))] =
[Q ∩ T1, CH(V (Q)), CH(V (Q))] = [V (Q), CH(V (Q))] = 1. Moreover, by (9.4),
|(Q ∩ T1)/CQ∩T1(T1)| = |V Q0/CV (V g )Q0| = |V/CV (V g )| = q.
Since H/CH(V (Q)) ∼ = SL2(q) is generated by two Sylow p-subgroups, therefore
(9.6)
|(Q ∩ T1)/D| ≤ q 2 and Q ∩ T1 = V D
for D := CQ∩T1(H). In particular, (b) holds. It remains to show that |D| ≤ q if (i),(ii)
or (iii) holds.
Note that by (9.6), (Q ∩ T1)/D is a natural SL2(q)-module for H/CH(V (Q)) and
therefore |CQ∩T1/D(T1)| = q. Also, DD g ≤ Z(T1) ≤ Q ∩ T1. Hence, if D ∩ D g = 1,
then
|D| = |D g | = |D g /D ∩ D g | = |D g D/D| ≤ q.
Thus, it remains to show that each of the conditions (i),(ii) and (iii) implies D∩D g = 1.
Assume first (i) holds. Then D ∩ D g is normal in T and T g . Moreover, D ∩ D g is
centralized by H and H g and hence F-characteristic in Q and Q g . Thus, 9.2.8 implies
together with (9.2) that D ∩ D g = 1, as required.
If (ii) holds then by (9.5), Q is elementary abelian. Hence, 3.4.6 implies |A(T )| =
2 and Q ∈ A(T ). Thus, if S = T then Q is normal in F, a contradiction. Hence, (ii)
implies (iii).
Therefore, we may assume from now on that (iii) holds. Then by (9.5), J(Q) =
Q ∩ T1. Since T �= S and p = 2, there is t ∈ NS(T )\T such that t 2 ∈ T . Since
T1 char T , we have T t 1 = T1. Note that by 8.3.2, NS(J(Q)) = T . Therefore,
(Q ∩ T1) t �≤ Q. By (9.6), Q ∩ T1 ≤ V Z(T1). Since Z(T1) t = Z(T1) ≤ Q, this yields
115

V t �≤ Q. Hence, we can replace g by t and may assume
(9.7)
g ∈ NG1(T ) and g 2 ∈ T.
Since T1 ✂ T , Q ∩ T1 ✂ T and T h 1 is normalized by Q for all h ∈ H. Hence, H =
Op′ (H) = 〈T h 1 | h ∈ H〉 is normalized by Q and therefore D ✂ Q. Now T = QT1 ≤
QH implies
D ✂ T.
In particular, by (9.7), (D ∩ D g ) g = D ∩ D g . Thus, (9.3) yields D ∩ D g = 1. This
completes the proof of Lemma 9.2.9.
9.3 The proof of Lemma 9.1.3
In this section we want to prove Lemma 9.1.3 and assume therefore that the converse
is true, i.e. that F is minimal and there is 1 �= U0 charF Q such that T < NS(U0).
Lemma 9.3.1. There is U ∈ C(Q) such that T < NS(U).
Proof. By 9.2.4, U := U ∗ 0 (Q) ∈ C(Q). Also, T < NS(U0) ≤ NS(U).
For the remainder of this section we set R := [V (Q), T ].
Lemma 9.3.2. (a) q = 2 = |R|.
(b) [V (Q), A ◦ (Q)] is a natural S3-module for A ◦ (Q)/CA ◦ (Q)(V (Q)).
(c) NNS(U)(R) = T for every U ∈ C(Q) which is of maximal order such that T <
NS(U).
Proof. By 9.3.1 there is U ∈ C(Q) such that T < NS(U). We choose U such that |U|
is maximal. We use the following notation:
R0 := [R, NA ◦ (Q)(TQ)],
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W = V (U),
T0 = NS(U),
T1 = T0 ∩ NS(T ).
Note that T < T1 since T ≤ T0. The following property follows from 9.2.4 and the
maximality of U.
(9.8)
We show next:
(9.9)
Let X charF Q such that U < X. Then NS(X) = T.
R ≤ ZCV (Q)(A ◦ (Q)) and R0 ≤ Z.
Let K = NA ◦ (Q)(TQ). Then Z is invariant under K and by 9.1.1, Z �≤ CR(A ◦ (Q)).
Therefore, it follows from the structure of V (Q) that R ≤ ZCV (Q)(A ◦ (Q)). Hence,
This proves (9.9).
R0 = [R, K] ≤ Z.
Set A∗(Y ) = A∗(Y, W ) for Y ≤ T . Property (I) implies J(Q) �≤ U. Hence we
get from (IV),
(9.10)
We show next:
∅ �= A∗(Q) ⊆ A∗(T ).
For all t ∈ T1\T there is A ∈ A∗(Q) with A t �≤ Q and A t (9.11)
∈ A∗(T ).
By (I), J(Q) �≤ U. Hence, U < X := 〈A∗(Q)〉U. Since X charF Q, it follows
from (9.8) that NS(X) = T . Thus, for t ∈ T1\T , X t �= X and therefore 〈A∗(Q)〉 t �=
〈A∗(Q)〉. Hence, there is A ∈ A∗(Q) with A t �∈ A∗(Q). Since A∗(T ) is invariant
under T1, it follows from 9.10 that A t ∈ A∗(T ) and A t �≤ Q. This shows (9.11).
Since T < T1, we can choose t ∈ T1\T . Then by (9.11), there is A ∈ A∗(Q)
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such that A t �≤ Q and A t ∈ A∗(T ). Now 9.2.6 yields W = V (Q)CW (A t ) and
R = [W, A t ] = [W, A] t . Hence, R t−1
= [W, A].
Note that by (9.9), [R, T ] = 1 = [Rt−1, T ] = [W, A, T ]. By 9.2.2(a), AU and thus
also [W, A] is O p (O p′
(NA ◦ (Q)(U)))-invariant. Therefore, it follows from the module
structure of V (Q) that [W, A] ∩ V (Q) ≤ CV (Q)(A ◦ (Q)). Hence, if R = R t then
R = R t−1
≤ C(A ◦ (Q)).
Hence, R �= R t and, since t ∈ T1\T was arbitrary, therefore NNT 0 (R)(T ) = NT1(R) =
T . Hence, NT0(R) = T and (c) holds. Moreover, by (9.9), R0 ≤ Rt−1 ∩ V (Q) ≤
C(A ◦ (Q)), and the module structure of V (Q) yields q = 2. In particular, |A t /(A t ∩
Q)| = 2 and hence, |[V (Q), A t ]| = |V (Q)/CV (Q)(A t )| = 2. Since A ◦ (Q) is generated
by CA ◦ (Q)(V (Q)) and two conjugates of A t , this implies |[V (Q), A ◦ (Q)]| ≤ 4, i.e. by
(III) [V (Q), A ◦ (Q)] ∼ = C2 × C2 and |R| = 2. This shows (a) and (b).
Lemma 9.3.3. Let 1 �= U charF Q. Then NS(R) ∩ NS(U) = T .
Proof. Set T0 = NS(U) and assume T < NT0(R). Then T < NT0(R) ∩ NT0(T ).
By 9.3.2, p = q = 2. Hence we can choose t ∈ NT0(R) ∩ NT0(T ) such that t �∈ T
and t 2 ∈ T . Then U ∗ (Q) ≤ U1 := Q ∩ Q t ✂ T 〈t〉 and by 9.2.3(e), U1 charF Q.
Moreover, |Q/U1| = |QQ t /Q t | = q = 2 and in particular, U1 = U ∗ 1 (Q). Applying
9.2.4 we get U1 ∈ C(Q) and T < NS(U1). Moreover |U1| is maximal with respect
to U1 having these two properties. Therefore, by 9.3.2(c), t ∈ NNS(U1)(R) = T , a
contradiction.
Definition 9.3.4. We define C ∗ (Q) to be the set of all 1 �= U charF Q such that
|U| = max{|U ∗ | : U ∗ charF Q, U ∗ ✂ NS(U)}.
Note that then U = U ∗ (Q) for every U ∈ C ∗ (Q). Hence, by 9.2.4, C ∗ (Q) ⊆ C(Q).
Lemma 9.3.5. Let U ∈ C ∗ (Q) and X ≤ Q such that X ✂ NS(U). Then X ≤ U.
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Proof. Since U charF Q, it follows from 9.2.3(e) that UX charF Q. Moreover,
UX ✂ NS(U). Hence the maximality of |U| yields X ≤ U.
Lemma 9.3.6. Let U ∈ C ∗ (Q) and X ≤ Q such that X �≤ U. Then there is t ∈ NS(U)
such that X t �≤ Q.
Proof. Otherwise 〈X NS(U) 〉 ≤ Q, a contradiction to 9.3.5 and X �≤ U.
Lemma 9.3.7. Let U ∈ C(Q), A ∈ A∗(T, V (U)) and b ∈ NS(U)\T such that A �≤ Q
and A b ≤ T . Then A b ≤ Q.
Proof. Assume A b �≤ Q. Then T = A b Q = AQ, and 9.2.6 implies V (U) =
V (Q)CV (U)(A) = V (Q)CV (U)(A b ). Hence,
R = [V (Q), A b ] = [V (U), A b ] = [V (U), A] b = [V (Q), A] b = R b .
This is a contradiction to 9.3.3.
Lemma 9.3.8. Let U ∈ C ∗ (Q). Then we have A∗(T, V (U)) = A∗(Q, V (U)) or
J(NS(U)) ≤ T .
Proof. Set T0 = NS(U), W = V (U) and A∗(Y ) = A∗(Y, W ) for every Y ≤ T0.
Assume J(T0) �≤ T and A∗(T ) �= A∗(Q). We show first:
(9.1)
There is B∗ ∈ A∗(T0) such that B∗ �≤ T.
By assumption, there is B∗ ∈ A(T0) with B∗ �≤ T . We may choose B∗ such that
|B∗U/U| is minimal. Let B ∈ A∗(B∗U). Then B ∈ A∗(T0). Let t ∈ T0 and observe
that B t ∈ A∗(T0). Assume (9.1) does not hold. Then B and B t are contained in T .
Suppose B t �≤ Q. Since B t ∗U/U is elementary abelian, B t U is normalized by B t ∗.
Hence, by 9.3.7, B t ∗ ≤ T . In particular, A(T ) ⊆ A(T0). Moreover, since T = B t Q,
we have Bt ∗U = Bt (Bt ∗U ∩ Q) = BtU(B t ∗ ∩ Q) and B∗U = BU(B∗ ∩ Qt−1). Note that B t ∗ �≤ Q, since B t �≤ Q. Hence, by (IV), C∗ = (B t ∗ ∩ Q)V (Q) ∈
119

A(T ) ⊆ A(T0). Therefore, Ct−1 ∗ = (B∗ ∩ Qt−1)V (Q) t−1 ∈ A(T0). As above, B∗U =
BU(B∗ ∩ Qt−1) = BUC t−1
∗ , BU ≤ T and B∗ �≤ T . Hence Ct−1 ∗
�≤ T . Therefore,
by the minimality of |B∗U/U|, Ct−1 ∗ U = B∗U. Then B∗ ≤ Qt−1, i.e. Bt ≤ Bt ∗ ≤ Q
which contradicts our assumption. Hence, B t ≤ Q.
Since t ∈ T0 was arbitrary we have shown that X = 〈B T0 〉 ≤ Q. Therefore it
follows from 9.3.5 that B ≤ X ≤ U, a contradiction to the choice of B. Thus, (9.1)
holds.
(9.2)
So we can choose now B∗ ∈ A∗(T0) such that B∗ �≤ T . Then 3.2.16 implies
B∗ acts quadratically on W.
Now let T ≤ Y ≤ T0 be maximal with respect to inclusion such that 〈A∗(Y )〉 ≤ T .
Then by (9.1), Y �= T0 and hence Y < NT0(Y ). So the maximality of Y implies
〈A∗(NT0(Y ))〉 �≤ T . Since 〈A∗(Y )〉 = 〈A∗(T )〉, we have NT0(Y ) ≤ NT0(〈A∗(T )〉).
Thus we can choose B∗ such that
(9.3)
We show next:
(9.4)
B∗ ≤ NT0(〈A∗(T )〉).
|B∗/CB∗(R)| = |B∗/NB∗(R)| = |B∗/B∗ ∩ T | = 2.
By 9.3.2(a) and 9.3.3, NB∗(R) = CB∗(R) = B∗ ∩ T and [R, B∗] �= 1. Let b ∈ B∗\T
and assume there is c ∈ B∗\((B∗ ∩T )∪b(B∗ ∩T )). Note that p = q = 2 and therefore
b and c are involutions. Moreover, b, c and bc are not elements of T . By assumption,
A∗(T ) �= A∗(Q), i.e. there is A ∈ A∗(T ) with A �≤ Q. Then by (9.3) and 9.3.7,
A b ≤ Q, A c ≤ Q and A cb ≤ Q.
In particular, [W0, A c ] = 1 for W0 = V (Q)V (Q) b . Observe that W0 ≤ V (Q)[V (Q), b],
and so by (9.2),
W0 = V (Q)CW0(B∗) = V (Q) b CW0(B∗).
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Moreover, [V (Q) b , A] = 1 and therefore [V (Q) bc , A c ] = 1. Since [W0, A c ] = 1, it fol-
lows [W c 0 , A c ] = [V (Q) bc CW0(B∗), A c ] = 1 and therefore [W0, A] = 1 = [V (Q), A].
This is a contradiction since A �≤ Q. Hence, (9.4) holds. We show now
(9.5)
CB∗(R) = CB∗(W ), |W/CW (B∗)| = |B∗/CB∗(W )| = 2, W = RCW (B∗).
It follows from (9.4) that |B∗/CB∗(R)| = 2 and [R, B∗] �= 1. Since B∗ is elementary
abelian, therefore R �≤ B∗ and |RCB∗(R)| = |B∗|. Observe that RCB∗(R) is ele-
mentary abelian, so RCB∗(R) ∈ A(T0). Since (RCB∗(R))U = CB∗(R)U is a proper
subset of B∗U, it follows from the minimality of B∗U that RCB∗(R) ≤ U. Then
CB∗(R) = CB∗(W ). Therefore, by 3.2.3 and (9.4),
1 �= |W/CW (B∗)| ≤ |B∗/CB∗(W )| = |B∗/CB∗(R)| = 2.
Hence, equality holds above and R �≤ CW (B∗) implies (9.5).
Now choose t ∈ A ◦ (Q)\NA ◦ (Q)(TQ)CA ◦ (Q)(V (Q)) and b ∈ B∗\CB∗(W ). Then
by (9.5), B∗ = CB∗(W )〈b〉. Set Y = RR t R b . Note that Y ≤ W , since R ≤ V (Q) ≤
W ∩ Ω(Z(Q)) and U charF Q. Using (9.5), we get [W, b] = [RCW (B∗), b] = [R, b] ≤
RR b ≤ Y . Hence,
Y b = Y.
As before let A ∈ A∗(T ) with A �≤ Q. Then by (9.3) and 9.3.7, A b ≤ Q. Hence,
[RR t , A b ] = [V (Q), A b ] = 1. Since [R, A] = [R, T ] = 1 = [R b , A b ], this implies
[Y, A b ] = 1. As we have shown above, Y = Y b . So we get 1 = [Y, A] b and hence
[Y, A] = 1. In particular, [R t , A] = 1 which is a contradiction to the module structure
of V (Q). This proves 9.3.8.
Lemma 9.3.9. Let 1 �= U charF Q. Then J(NS(U)) ≤ T .
Proof. Assume the assertion is wrong. Then we can choose U of maximal order such
that U charF Q and J(NS(U)) �≤ T . Set T0 := NS(U). Note that U ∈ C ∗ (Q). Thus,
121

y 9.3.8, A∗(T, V (U)) = A∗(Q, V (U)). Set X := 〈A∗(Q, V (U))〉. Then X �≤ U
and U1 := XU charF Q. Therefore, by the choice of U, J(NS(U1)) ≤ T ≤ T0. In
particular, J(T ) = J(NS(U1)) = J(NT0(U1)) and
A∗(Q, V (U)) = A∗(T, V (U)) = A∗(NT0(U1), V (U)).
Hence, NT0(NT0(U1)) normalizes XU = U1. It follows that T0 ≤ NS(U1), which
contradicts the maximality of U.
Lemma 9.3.10. Let U charF Q. Then B(NS(U)) = B(T ).
Proof. Set T0 := NS(U). By 9.3.9, J(T0) = J(T ) and so B(T0) = CT0(Ω(Z(J(T )))).
Since R ≤ Ω(Z(T )), we have R ≤ Ω(Z(J(T ))). Hence, by 9.3.3, B(T0) ≤ NT0(R) =
T . This shows B(T0) = B(T ).
Now set T1 = B(T ).
Lemma 9.3.11. T1 ∩ Q is elementary abelian and |T1 ∩ Q| ≤ 2 3 . Furthermore,
there is H ≤ M such that T1 ∈ Sylp(H), H = Op′ (H), M = HCM(V (Q)) and
(T1 ∩ Q)/CT1∩Q(H) is a natural S3-module for H/CH(V (Q)).
Proof. Let 1 �= U charF Q such that NS(U) is maximal with respect to inclusion.
We may also choose U such that U is maximal with respect to these properties. Then
U ∈ C ∗ (Q). Set T0 = NS(U).
It follows from (I)-(III) that M fulfils the hypothesis of 4.2.2. Hence, there is
H ≤ M such that M = CM(V (Q))H and T1 ∈ Sylp(H). We choose H minimal with
this property. Note that then H = Op′ (H).
By (IV), we have V (Q) ≤ J(Q) ≤ J(T ) ≤ T1. Moreover, 9.3.10 implies
T1 char T0. For 1 �= U0 ≤ T1 ∩ Q with U0 ✂ T0 and U0 charF Q, it follows from the
choice of U that NS(U0) = T0. Therefore, the hypothesis of 9.2.9 holds. Hence, by
9.2.9(a), H has the pushing up property and Op(H) = Q ∩ T1.
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Assume there is a characteristic subgroup C of T1 with H = CH(V (Q))NH(C).
Then T1 ∈ Sylp(NH(C)) and M = CM(V (Q))NH(C). Since H is minimal, it follows
C ✂ H, a contradiction since H has the pushing up property. Therefore Hypothesis
4.1.2 holds for H and T1 instead of G and T . Then by Corollary 4.1.3, there is a
subgroup H ∗ of H such that T1 ≤ H ∗ , H = CH(V (Q))H ∗ and [Op(H), O p (H ∗ )] ≤
V (Q). Then by the minimal choice of H, H = H ∗ and [Q ∩ T1, O p (H)] ≤ V (Q).
Moreover, T1 char T and by assumption, T �= S. Hence, 9.2.9(b) implies the asser-
tion.
The final contradiction. By 9.3.11, Q∩T1 is elementary abelian of order at most 2 3 , and
there is H ≤ M such that M = HCM(V (Q)), T1 ∈ Sylp(H), H = Op′ (H) and Q ∩
T1/CQ∩T1(H) is a natural S3-module for H/CH(V (Q)). In particular, |CQ∩T1(H)| ≤
2.
Note that Z ≤ J(T ) ≤ T1 and by 9.1.2, Z �≤ Z(H). Therefore, the module
structure of Q ∩ T1 implies Ω(Z(T1)) ≤ ZCQ∩T1(H).
Observe that [T1, Q] ≤ T1 ∩ Q = Op(H) and H = Op′ (H) = 〈T H 1 〉. Hence,
[Q, H] ≤ Op(H) ≤ H. In particular, Q normalizes and therefore also centralizes
CT1∩Q(H), since |CQ∩T1(H)| ≤ 2. Thus, CQ∩T1(H) ≤ Z(T ), since T = T1Q. This
shows Ω(Z(T1)) ≤ ZCQ∩T1(H) ≤ Z(T ). By the definition of T1, Ω(Z(J(T ))) ≤
Ω(Z(T1)) and hence, T = B(T ) = T1.
Thus, Q = Q ∩ T1 is elementary abelian. By 9.3.1, we can choose U ∈ C(Q)
such that T < NS(U). Then for t ∈ (NS(T ) ∩ NS(U))\T , T = QQ t and U =
U t ≤ Q ∩ Q t ≤ Z(T ). Since Z ≤ U and U charF Q, we have V (Q) ≤ U. Hence,
V (Q) ≤ Z(T ), a contradiction.
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9.4 The proof of Corollary 9.1.4
Assume F is minimal and there is a non-trivial characteristic subgroup C of T such
that M = CM(V (Q))NM(C). Then C charF Q and C ✂ NS(T ). Hence, by Lemma
9.1.3, NS(T ) = T , i.e. S = T . Therefore C is A(S)-invariant. Now 9.2.7 yields
C = 1, a contradiction.
9.5 The proof of Theorem 2
Lemma 9.5.1. Let F be minimal. Then the following hold:
(a) Let 1 �= U charF Q. Then U is not A(T )-invariant.
(b) Let T ≤ H ≤ M such that M = CM(V (Q))H. Then there is x ∈ NG1(T ) such
that [Q, O p (H)] x �≤ Q. If T < S, we may choose x such that x ∈ NS(T ).
Proof. Assume there is 1 �= U charF Q such that U is A(T )-invariant. Then U ✂
NS(T ) and by Lemma 9.1.3, NS(T ) = T , i.e. S = T . Hence, U is A(S)-invariant, a
contradiction to 9.2.7. Thus, (a) holds.
Assume now (b) does not hold, i.e. U := 〈[Q, O p (H)] NG 1 (T ) 〉 ≤ Q. Then
[U, O p (H)] ≤ [Q, O p (H)] ≤ U and U ✂ H = T O p (H). Therefore, U charF Q.
Since 1 �= U is A(T )-invariant, this contradicts (a).
The proof of Theorem 2. Let F be minimal. It follows from Corollary 9.1.4 that M
fulfils Hypothesis 4.1.2. Therefore, by Corollary 4.1.3, there is a subgroup H of M
such that T ≤ H, M = CM(Ω(Z(Q)))H and, for V := [Q, O p (H)], either
(I’) V ≤ V (Q) and V/CV (H) is a natural SL2(q)-module for H/CH(V (Q)), or
(II’) Z(V ) ≤ Z(Q), p = 3, V/Z(V ) and Z(V )/Φ(V ) are natural SL2(q)-modules
for H/CH(V (Q)), and Φ(V ) = CV (M) has order q.
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Moreover, 4.1.3 implies the following.
(i) Q = V CQ(L) for some subgroup L of H with H = LQ.
(ii) If (II’) holds then Qx2 = Q for every x ∈ NG1(T ) with V x �≤ Q.
(iii) Φ(D) x = Φ(D) for D = CQ(O p (H)) and every x ∈ A(T ) with V x �≤ Q.
Thus it is sufficient to show that (I’) implies (I) of Theorem 2 and (II’) implies (II) of
Theorem 2.
If (I’) holds, then it follows from 9.2.9 applied with T1 = T0 = T that Q is
elementary abelian. Thus, (I) of Theorem 2 follows from Theorem 1. Moreover, again
by 9.2.9, |Q| ≤ q 3 if p = 2 or F is strongly minimal.
Assume now (II’) holds. By (i), there is L ≤ H such that H = LQ and Q =
V CQ(L). Then O p (H) = O p (L) ≤ L. Hence, Q = V D. By 9.5.1, there is x ∈
NG1(T ) such that V x �≤ Q.
Since Φ(D) charF Q it follows from (iii), 9.2.8 and the choice of x that Φ(D) = 1.
Furthermore, [V, D] = 1 and Q = V D. Hence, D ≤ Ω(Z(Q)) and thus D = CQ(M).
Therefore, D is normal in G2. In particular, by (ii), Dx2 = D. Set U := D ∩ Dx . Then
U x = U and U charF Q. Hence, 9.2.8 and the choice of x yields U = 1.
Note that D = CQ(M) ≤ Z(T ) and thus also D x ≤ Z(T ) ≤ Q. Since Q =
CD, we have Q/D ∼ = V/CV (O p (H)) = V/CV (H). So it follows from 3.4.2 that
|CQ/D(V x )| ≤ q 2 . Hence,
|D| = |D x | = |D x /D ∩ D x | = |D x D/D| ≤ |CQ/D(V x )| ≤ q 2 .
This yields |Q| = |V D| = |V/(V ∩ D)| |D| = |V/Φ(V )| |D| ≤ q 4 · q 2 = q 6 .
It remains to show that T = S. Assume T < S. Then by 9.5.1, we may choose
x ∈ NS(T ) such that V x �≤ Q. We show first
125

(iv) x 2 ∈ NG1(Q) for every x ∈ NG1(T ).
Let x ∈ NG1(T ). We may assume that Q x �= Q. If V = V x , then [V, Q] ≤ Φ(V )
implies [V, Q x ] = [V x , Q x ] ≤ Φ(V x ) = Φ(V ), a contradiction to the module structure
of V . Thus, V �= V x .
If V x �≤ Q, then (iv) follows from (ii). Thus, we may assume V x ≤ Q and Q x �= Q.
Since NS(J(Q)) = T , it follows from 3.2.3 and 3.4.3 that |Q/Q ∩ Q x | = |Q x Q/Q| =
q. Thus, |Q| ≤ q 6 implies |Q ∩ Q x | ≤ q 5 . Since V x ≤ Q ∩ Q x and |V x | = |V | = q 5 ,
V x = Q ∩ Q x . If V ≤ Q x , then V ≤ Q ∩ Q x = V x and thus V = V x , a contradiction.
Hence V �≤ Q x and therefore V x−1
Q x2
�≤ Q. Thus, by (ii), Q x−2
= Q. This yields
= Q and shows (iv). In particular, x ∈ NG1(Q) for every x ∈ NG1(T ) of odd
order. Thus, NS(T ) = T and hence T = S.
126

Chapter 10
A Classification for p = 2
Let F be a saturated fusion system on a finite 2-group S. Suppose that F is minimal
and of essential rank 1. Let Q ≤ S be essential and let A = (G1, G2, G12, φ1, φ2) be
the amalgam constructed in Proposition 1. Set
T = NS(Q),
M = O p′
(G2),
H = G1 ∗G12 G2.
We will from now on consider G1 and G2 as subgroups of H. In particular we identify
NG1(Q) with NG2(T ).
10.1 Bounding the 2-structure
Remark 10.1.1. Q is elementary abelian and |Q| ≤ q 3 . Moreover, M/Q ∼ = SL2(q)
and Q/CQ(M) is a natural SL2(q)-module for M/Q.
Proof. This is a consequence of Theorem 2.
From now on we will use 10.1.1 without reference.
Lemma 10.1.2. There is x ∈ NG1(T ) with Q x �= Q.
Proof. We need to show that NG1(Q) < NG1(T ). Note that Q is not normal in G1,
since Q is not normal in F. Thus we may assume that T is not normal in G1. Then by
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the choice of G1, S �= T . Hence, T < NS(T ) and again the assertion follows.
From now on we choose x ∈ NG1(T ) with Q x �= Q.
Lemma 10.1.3. Q ∈ A(T ), T = QQ x , |Q/(Q ∩ Q x )| = q and CQ(a) = Q ∩ Q x =
Z(T ) for a ∈ T \Q.
Proof. This follows from 3.4.5 and the choice of x.
Lemma 10.1.4. There is E ≤ G2 such that G2 is the semidirect product of Q and E,
E ∩ M ∼ = SL2(q) and Q/CQ(E ∩ M) is a natural SL2(q)-module for E ∩ M.
Proof. By 10.1.3, there is a complement of Q in T . Since Q is an abelian normal
subgroup of G2 and |Q| is coprime to |G2 : T |, we can apply a Theorem of Gaschütz
(see for example [15, 3.3.2]). This gives us the existence of a complement E of Q
in G2. Then M = (E ∩ M) ⋉ Q, E ∩ M ∼ = M/Q ∼ = SL2(q) and CQ(M) =
CQ(E ∩ M).
From now on we fix E with the properties described in 10.1.4. Then T = Q(T ∩ E).
Also, Q ∩ E = 1, |T ∩ E| = q and T ∩ E ∈ Sylp(E ∩ M). Since E ∩ M ∼ = SL2(q),
we can choose d ∈ NE∩M(T ∩ E) of order q − 1.
Lemma 10.1.5. Suppose x ∈ NS(T )\T and [x, d] ≤ T . Then |Q| = q 2 or q = 2.
Proof. Assume q �= 2. Then d �= 1 and C := CT (d) = CQ(d) = CQ(M). Since d
acts coprimely on T1 := NS(T ), our assumption implies C < CT1(d) and therefore
C < NT1(C) ∩ CT1(d). However, CT1(d) ∩ T = C. Now 9.1.3 yields C = 1 and
therefore |Q| = q 2 .
Lemma 10.1.6. Suppose x ∈ NS(T )\T and [x, d] ≤ T . Then q ≤ 4.
Proof. We may assume that q �= 2. Then by 10.1.5, Q is a natural SL2(q)-module for
M/Q. In particular, d acts fixed point freely on T . The coprime action of d on T 〈x〉
gives now T 〈x〉 = T × CT 〈x〉(d). Hence, we may assume that [x, d] = 1. By 10.1.3,
we can apply 3.4.7 and get q ≤ 4.
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Lemma 10.1.7. A(T ) = {Q, Q x } and Q # ∪ (Q x ) # is precisely the set of involutions
in T . Moreover, |NS(T )/T | ≤ 2.
Proof. This follows from 10.1.3 and 3.4.6.
Lemma 10.1.8. If J(NS(T )) �≤ T , then q = 2 and |Q| = q 2 .
Proof. Set T1 = NS(T ) and assume J(T1) �≤ T . Then there is B ∈ A(T1) such that
B �≤ T and we may choose x such that x ∈ B\T . By 10.1.7, |B/(B ∩T )| = |T1/T | =
2. Since B ∩ T is elementary abelian, it follows from 10.1.7 that every element of
B ∩ T is contained in Q or Q x . Since B ∩ T is centralized by x, B ∩ T ≤ Q ∩ Q x .
Also note that |Q| ≤ |B|, since B ∈ A(T1). Therefore, q = |T/Q| = |Q/(Q ∩ Q x )| ≤
|Q/(T ∩ B)| ≤ |B/(T ∩ B)| = 2. Thus, q = 2 and U := CQ(M) ≤ Q ∩ Q x = B ∩ T .
Hence, [U, B] = 1 and by Theorem 9.1.3, U = 1. This shows the assertion.
Lemma 10.1.9. If J(NS(T )) ≤ T then |S/T | = 2 and G1 = SNG1(Q).
Proof. Set T1 = NS(T ) and assume J(T1) ≤ T . Then by 10.1.3, J(T1) = J(T ) = T .
Hence, T ✂ NS(T1) and therefore NS(T1) = T1. This yields S = T1 and T = J(S) ✂
G1. In particular, by 10.1.7, |S/T | ≤ 2.
Now let y ∈ G1 be of odd order. Note that y acts on A(T ), Q ∈ A(T ) and by
10.1.7, |A(T )| = 2. Hence, Q y2
= Q. Since y has odd order, this yields Q y = Q.
Hence, y ∈ NG1(Q) for every y ∈ G1 of odd order. Since G1 is p-closed we get
G1 = SNG1(Q). Now note that Q is not normal in F and hence also not normal in G1.
Thus, S �= T and |S/T | = 2.
Lemma 10.1.10. If q �= 2 then |Q| = q 2 and q = 4.
Proof. By 10.1.9, S �= T . Thus we may choose x such that x ∈ NS(T )\T . Note that d
acts on S and T and therefore on NS(T )/T . By 10.1.7, |NS(T )/T | ≤ 2 and it follows
that [x, d] ≤ T . Now the assertion follows from 10.1.5 and 10.1.6.
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10.2 Identifying F
Lemma 10.2.1. We have G2 = M. Moreover, if J(NS(T )) ≤ T , then G1 = S〈d〉.
Proof. By 10.1.10, M/Q ∼ = SL2(q) where q ∈ {2, 4}. Therefore, by 2.6.7, every
p ′ -automorphism of M/Q is an inner automorphism. Hence, G2 = MCG2(M/Q).
We assume now M �= G2. Then there is y ∈ CG2(M/Q) such that y �∈ M.
Since G2/M is a p ′ -group, we can assume that y has p ′ -order. By the choice of y, in
particular, y ∈ CG2(T/Q) ≤ NG2(T ) ≤ G1. Therefore, y normalizes S and T1 :=
NS(T ). Moreover, by 10.1.7, |T1/T | ≤ 2. Hence, [T1, y] ≤ T . It follows now from
2.3.1 that CT1(y) �≤ T , i.e. we may choose x ∈ T1\T such that [x, y] = 1. Then [T, y]
is normalized by x and [T, y] ≤ Q ∩ Q x = Z(T ). In particular, [Q, y] ≤ Z(T ). Thus,
again by 2.3.1, we have CQ(y) �= 1.
Now note that [M, y] ≤ Q and y acts on Q as an element of F := End
GF (p)
M
(Q).
By Schur’s Lemma and Wedderburn’s Theorem, F is a field and Q is an F -vector
space. Hence, it follows from CQ(y) �= 1 that [Q, y] = 1, i.e. y ∈ CG2(Q) ≤ Q. This
is a contradiction to the choice of y and shows M = G2. In particular, NG1(Q) =
NG2(T ) = T 〈d〉. If J(NS(T )) �≤ T it follows now from 10.1.9 that G1 = S〈d〉.
Theorem 10.2.2 (Classification of F if p = 2). One of the following cases holds:
(a) q = 2, |Q| = 4, G1 = S, G2 ∼ = S4, G12 ∼ = D8 and S is dihedral of order 2 n for
some n ≥ 4.
(b) q = 2, |Q| = 4, G1 = S, G2 ∼ = S4, G12 ∼ = D8 and S is semidihedral of order 2 n
for some n ≥ 4.
(c) q = 2, |Q| = 2 3 , |S| = 2 5 , G1 = S, G12 ∼ = C2 × D8 and G2 ∼ = C2 × S4.
(d) q = 4, |Q| = q 2 , |S| = 2 7 . Then G1 is the semidirect product of a cyclic group
of order 3 with S and G2 is the semidirect product of a subgroup E ∼ = SL2(4)
130

and Q such that Q is a natural SL2(4)-module for E.
Moreover, in each of the four cases, F is uniquely determined up to isomorphism.
Proof. It is sufficient to show that always one of the cases (a)-(d) holds and in each case
A is uniquely determined up to isomorphism. Then by 6.1.1, F is uniquely determined
up to isomorphism.
Assume first q = 2 and |Q| = q 2 . Then by 2.4.5, (a) or (b) holds. Moreover, by
2.9.3, the amalgam A is in both cases uniquely determined up to isomorphism.
Suppose now q �= 2 or |Q| �= q 2 . Then by 10.1.8, J(NS(T )) ≤ T . Thus, 10.1.9
implies |S/T | = 2.
If q �= 2, then by 10.1.10, q = 4 and |Q| = q 2 . In particular, |S| = |T | · 2 =
q 3 ·2 = 2 7 . Now (d) follows from 10.1.4 and 10.2.1. Moreover, by 2.9.6, A is uniquely
determined up to isomorphism.
Therefore we can assume that q = 2 and |Q| �= q 2 . Then by 10.1.1, |Q| = 2 3 .
Since |S/T | = 2, |S| = 2 5 . By 10.1.4 and 10.2.1, S = G1 and there is a subgroup E
of G2 such that E ∼ = S3 and G2 = EQ. Then Q = [Q, E] × CQ(E) and [Q, E] is a
natural S3-module for E. This shows G2 ∼ = S4 × C2. Since T ∈ Syl2(G2), it follows
that G12 ∼ = NG2(T ) = T ∼ = C2 × D8. This proves (c). Furthermore, by 9.1.3, we
can apply 2.9.5 and get that A is uniquely determined up to isomorphism. Hence the
assertion holds.
Proof of Theorem 3. In Chapter 7 we have shown that the fusion systems listed in
Theorem 3 are actually minimal of essential rank 1. Now the assertion follows from
Theorem 10.2.2.
131

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