SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

More documents

Recommendations

Info

$Øvelse 1: Få LATEX op at stå Øvelse 2: At skrive matematiske ...$

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
Page 1: SATURATED FUSION SYSTEMS OF ESSENTI
Page 5 and 6: 4.2 The Baumann subgroup . . . . .
Page 7 and 8: Chapter 1 Introduction Let G be a f
Page 9 and 10: fusion systems. A significant step
Page 11 and 12: 1 to be the first interesting case.
Page 13 and 14: We call F strongly minimal if Op(F)
Page 15 and 16: Even if there is no direct relation
Page 17 and 18: • Op′ (G) is the smallest norma
Page 19 and 20: 2.2 Commutators For elements x, y
Page 21 and 22: to apply the results about coprime
Page 23 and 24: are the only involutions in Aut(X).
Page 25 and 26: Let x, t ∈ D1 such that t is an i
Page 27 and 28: on 〈y〉. Since y c = y 5 , y j c
Page 29 and 30: matrix A ∈ GLn−1(q) we set ⎛
Page 31 and 32: Lemma 2.6.2. (a) Sylp(SL(V )) = Syl
Page 33 and 34: Proof. If q ∈ {2, 3} then |T ∩
Page 35 and 36: Proof. Let V be a 2-dimensional vec
Page 37 and 38: H is strongly p-embedded in G. Thus
Page 39 and 40: Remark 2.9.2. Suppose H := G12 = ˜
Page 41 and 42: Proof. Let A be the group consistin
Page 43 and 44: Observe that, in particular, no ele
Page 45 and 46: semilinear with respect to FA and F
Page 47 and 48: Now equality holds above, i.e |A/CA
Page 49 and 50: We will use 3.2.9 and 3.2.10 most o
Page 51 and 52: 3.3 Natural Sn-modules Definition 3
Page 53 and 54:
Lemma 3.4.3. Let G ∼ = SL2(q) and
Page 55 and 56:
is a group isomorphism. For µ ∈
Page 57 and 58:
Theorem 3.5.4. Assume Hypothesis 3.
Page 59 and 60:
O2(G) = 1. Hence, H = E and A ≤ E
Page 61 and 62:
subgroup E of G with E ∼ = SL2(q)
Page 63 and 64:
Chapter 4 Pushing Up Let H be a fin
Page 65 and 66:
(II) Z(V ) ≤ Z(Op(G)), p = 3, and
Page 67 and 68:
• V/CV (H) is a natural SL2(q)-mo
Page 69 and 70:
(3) For each φ ∈ MorF(P, Q), φ|
Page 71 and 72:
are bijective. Moreover, together w
Page 73 and 74:
(I) If P is fully normalized, then
Page 75 and 76:
5.3 Normalizers Definition 5.3.1. L
Page 77 and 78:
Lemma 5.4.3. Let U ≤ R ≤ S, N =
Page 79 and 80:
Note that for Q ∈ F, every elemen
Page 81 and 82:
We set R0 = R, Ri = Ri−1α ∗∗
Page 83 and 84:
normalizes AutS(P ) which is a Sylo
Page 85 and 86:
Lemma 5.7.3. Let F be of essential
Page 87 and 88:
Proof. Set N = NF(U), N = N /U and
Page 89 and 90:
such that G1α = H1 and G2α = H2.
Page 91 and 92:
NA(Q)(AutS(Q)) ≤ RF(Q). On the ot
Page 93 and 94:
elementary abelian subgroups of S
Page 95 and 96:
subgroups of NF(Q). Assume first th
Page 97 and 98:
(c) Let 1 �= W < V . Then NG(W )
Page 99 and 100:
Chapter 8 Recognizing SL2(q) in Fus
Page 101 and 102:
(a) CA(Q)(V )/Inn(Q) is a p ′ -gr
Page 103 and 104:
Lemma 8.2.7. 1 Let U ∈ C0(Q), set
Page 105 and 106:
C P (Z) ≤ N P (Z). Therefore we c
Page 107 and 108:
8.3 The proof of Theorem 8.1.1 From
Page 109 and 110:
M := Op′ (G2). By the constructio
Page 111 and 112:
is invariant under NA(S)(U) and the
Page 113 and 114:
Set U ◦ = U1U2. Since U and Z are
Page 115 and 116:
In particular, (a) holds. To ease n
Page 117 and 118:
V t �≤ Q. Hence, we can replace
Page 119 and 120:
such that A t �≤ Q and A t ∈
Page 121 and 122:
A(T ) ⊆ A(T0). Therefore, Ct−1
Page 123 and 124:
y 9.3.8, A∗(T, V (U)) = A∗(Q, V
Page 125 and 126:
9.4 The proof of Corollary 9.1.4 As
Page 127 and 128:
(iv) x 2 ∈ NG1(Q) for every x ∈
Page 129 and 130:
the choice of G1, S �= T . Hence,
Page 131 and 132:
10.2 Identifying F Lemma 10.2.1. We
Page 133 and 134:
Bibliography [1] J.L.Alperin, Sylow
show all

SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?