Fusion systems and amalgams

Andersen, Kasper K. S.; Oliver, Bob; Ventura, Joana

doi:10.1007/s00209-012-1109-6

Cited by 12 publications

(20 citation statements)

References 30 publications

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“…Points (a) and (b) follow from [AOV2,Proposition 5.2(a,b)], (c) follows from [AKO,Corollary I.8.5], and (d) and (h) from Lemma 2.1. Since F -essential subgroups are critical, point (e) follows from Lemma 1.16.…”

Section: Computer Search Criteriamentioning

confidence: 99%

“…If S is dihedral, semidihedral, or a wreath product C 2 n ≀ C 2 , then by [AOV1,§ 4.1] or [AOV2, Proposition 3.1], F is isomorphic to the fusion system of PSL 2 (q) or PSL 3 (q) for appropriate odd q, and we are in one of the cases (a), (b), or (c). If not, then |S| = 64 by [AOV2,Theorem 5.3], and so S is isomorphic to UT 4 (2) or UT 3 (4) or is of type M 12 by [AOV2,Theorem 5.4]. The reduced fusion systems over these three groups are listed in [O2,Propositions 5.1,6.4,& 4.2], and we are in the situation of (d), (e), or (f).…”

Section: -Groups Of Order At Most 128mentioning

confidence: 99%

“…When F = F S (G) for G = M 12 , then by [O2 , Proposition 4.3(a,b)], the composite µ G • κ G is an isomorphism from Out (G) to Out (F ). By [AOV2, Proposition 3.2], there are at most two F -essential subgroups, of which only one (R in the notation of [AOV2] and A + in [O2, § 4]) has noncyclic center. Hence Ker(µ G ) = 1 by Proposition 1.13, κ G is an isomorphism, and F is tame.…”

Section: -Groups Of Order At Most 128mentioning

confidence: 99%

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Reduced fusion systems over 2-groups of small order

Andersen

Oliver

Ventura³

2017

Journal of Algebra

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Abstract. We prove, when S is a 2-group of order at most 2 9 , that each reduced fusion system over S is the fusion system of a finite simple group and is tame. It then follows that each saturated fusion system over a 2-group of order at most 2 9 is realizable. What is most interesting about this result is the method of proof: we show that among 2-groups with order in this range, the ones which can be Sylow 2-subgroups of finite simple groups are almost completely determined by criteria based on Bender's classification of groups with strongly 2-embedded subgroups.A saturated fusion system over a finite p-group S is a category whose objects are the subgroups of S, whose morphisms are monomorphisms between subgroups, and which satisfy certain axioms first formulated by Puig [Pg] and motivated in part by conjugacy relations among p-subgroups of a given finite group. A saturated fusion system is realizable if it is isomorphic to the fusion system defined by the conjugation relations within a Sylow p-subgroup of some finite group, and is exotic otherwise. One of our main goals is to try to understand when and how exotic fusion systems can occur, especially over 2-groups.A saturated fusion system F is reduced if O p (F ) = 1 and O p (F ) = O p ′ (F ) = F (see Definitions 1.1(c,e) and 1.9(a)). A saturated fusion system F is tame if it is realized by a group G such that the natural homomorphism from Out(G) to a certain group of outer automorphisms of F (more precisely, of an associated linking system) is split surjective (Definition 1.10). The main result in our earlier paper [AOV1] says roughly that exotic fusion systems can be detected via tameness of associated reduced fusion systems. More precisely, by [AOV1, Theorems A & B], if the "reduction" of a fusion system F is tame, then F is tame and hence realizable, while if a reduced fusion system is not tame, then it is the reduction of an exotic fusion system.A saturated fusion system is indecomposable if it does not split as a product of fusion systems over nontrivial p-groups. We can now state our main result.Theorem A. Let F be a reduced, indecomposable fusion system over a nontrivial 2-group of order at most 2 9 . Then F is the fusion system of a finite simple group, and is tame.Proof. This is shown in Theorems 4.1 (for 2-groups of order at most 64), 4.3 (order 2 7 ), 5.1 (order 2 8 ), and 6.1 (order 2 9 ).The next theorem follows from Theorem A and the above discussion.Theorem B. Each saturated fusion system over a 2-group of order at most 2 9 is realizable.

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Section: Computer Search Criteriamentioning

confidence: 99%

Section: -Groups Of Order At Most 128mentioning

confidence: 99%

Section: -Groups Of Order At Most 128mentioning

confidence: 99%

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Reduced fusion systems over 2-groups of small order

Andersen

Oliver

Ventura³

2017

Journal of Algebra

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“…When S ∈ DSW, F is as described in (1)-(3) by [AOV1,Propositions 4.3 & 4.4] and [AOV2,Proposition 3.1]. When S ∈ G, F is as in (4) by Proposition 4.2; and when S ∈ V (cases (5)- 7, and including the case S ∼ = UT 4 (2)) by Propositions 5.1, 5.5, and 5.6.…”

Section: Introductionmentioning

confidence: 99%

Reduced fusion systems over 2-groups of sectional rank at most 4

Oliver

2016

Memoirs of the AMS

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Chapter 1. Background on fusion systems 1.1. Essential subgroups in fusion systems 1.2. Reduced fusion systems 1.3. The focal subgroup Chapter 2. Normal dihedral and quaternion subgroups Chapter 3. Essential subgroups in 2-groups of sectional rank at most 4 3.1. Essential subgroups of index 4 in their normalizer 3.2. Essential pairs of type (II) 3.3. Essential subgroups of index 2 in S Chapter 4. Fusion systems over 2-groups of type G 2 (q) Chapter 5. Dihedral and semidihedral wreath products Chapter 6. Fusion systems over extensions of UT 3 (4) Appendix A. Background results about groups Appendix B. Subgroups of 2-groups of sectional rank 4 Appendix C. Some explicit 2-groups of sectional rank 4 Appendix D. Actions on 2-groups of sectional rank at most 4

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“…Stancu considered the saturated fusion systems over the generalized extra special p-groups and metacyclic p-groups of odd order in [25,26]. More classification work can be found in [1,2,10,[15][16][17]. Several families of exotic fusion systems were constructed in [7][8][9]14,21].…”

Section: Introductionmentioning

confidence: 99%