SATURATED FUSION SYSTEMS OF ESSENTIAL RANK 1

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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 normalizers 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 subgroups. 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
Page 1 and 2: SATURATED FUSION SYSTEMS OF ESSENTI
Page 3 and 4: Acknowledgements First of all I wou
Page 5 and 6: 4.2 The Baumann subgroup . . . . .
Page 7 and 8: Chapter 1 Introduction Let G be a f
Page 9: fusion systems. A significant step
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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