SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

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.
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