SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

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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
Page 1 and 2: SATURATED FUSION SYSTEMS OF ESSENTI
Page 3: Acknowledgements First of all I wou
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
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(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
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SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

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