SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

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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 published 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 subgroups 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
Page 1 and 2: SATURATED FUSION SYSTEMS OF ESSENTI
Page 3 and 4: Acknowledgements First of all I wou
Page 5: 4.2 The Baumann subgroup . . . . .
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.
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O2(G) = 1. Hence, H = E and A ≤ E
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subgroup E of G with E ∼ = SL2(q)
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Chapter 4 Pushing Up Let H be a fin
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(II) Z(V ) ≤ Z(Op(G)), p = 3, and
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• V/CV (H) is a natural SL2(q)-mo
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(3) For each φ ∈ MorF(P, Q), φ|
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are bijective. Moreover, together w
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(I) If P is fully normalized, then
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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
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We set R0 = R, Ri = Ri−1α ∗∗
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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
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such that G1α = H1 and G2α = H2.
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NA(Q)(AutS(Q)) ≤ RF(Q). On the ot
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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
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(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
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is invariant under NA(S)(U) and the
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Set U ◦ = U1U2. Since U and Z are
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In particular, (a) holds. To ease n
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V t �≤ Q. Hence, we can replace
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such that A t �≤ Q and A t ∈
Page 121 and 122:
A(T ) ⊆ A(T0). Therefore, Ct−1
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y 9.3.8, A∗(T, V (U)) = A∗(Q, V
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9.4 The proof of Corollary 9.1.4 As
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(iv) x 2 ∈ NG1(Q) for every x ∈
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the choice of G1, S �= T . Hence,
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10.2 Identifying F Lemma 10.2.1. We
Page 133 and 134:
Bibliography [1] J.L.Alperin, Sylow
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