Fusion Systems in Algebra and Topology

Aschbacher, Michael; Kessar, Radha; Oliver, Bob

doi:10.1017/cbo9781139003841

Cited by 225 publications

(633 citation statements)

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“…A fusion system is saturated if it satisfies certain additional conditions. Rather than listing those conditions here, we refer to [AKO,Definition I.2.2] or our earlier paper [AOV].…”

Section: Saturated Fusion Systemsmentioning

confidence: 99%

“…[AKO,Theorem I.2.3]). An abstract fusion system F over S is called realizable if F = F S (G) for some finite group G with S ∈ Syl p (G), and is called exotic otherwise.…”

Section: Saturated Fusion Systemsmentioning

confidence: 99%

“…We refer to [AKO,Proposition A.7] for a very brief survey of some of the properties of strongly p-embedded subgroups, and to [A1, § 46] or [Sz2,§ 6.4] for more details. Definition 1.…”

Section: Saturated Fusion Systemsmentioning

confidence: 99%

“…See, e.g., [O1, Proposition 1.10(a,b)]. In fact, a proper subgroup P < S fully normalized in F is F-essential exactly when Aut F (P ) is not generated by restrictions of morphisms between strictly larger subgroups of S. (See [OV,Proposition 2.5] or [AKO,Proposition I.3.3(b)] for more details. ) The next proposition follows easily from Theorem 1.2, together with the definition of a normal p-subgroup.…”

Section: Saturated Fusion Systemsmentioning

confidence: 99%

“…See, e.g., [AKO,Corollary I.7.5]. The second equivalence follows upon checking that the image of foc(F) in S/hyp(F) is precisely its commutator subgroup (cf.…”

Section: (B) [G θ] ≤ Fr(p )mentioning

confidence: 99%

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Fusion systems and amalgams

Andersen

Oliver

Ventura³

2012

Math. Z.

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Abstract. We study reduced fusion systems from the point of view of their essential subgroups, using the classification by Goldschmidt and Fan of amalgams of prime index to analyze certain pairs of such subgroups. Our results are applied here to study reduced fusion systems over 2-groups of order at most 64, and also reduced fusion systems over 2-groups having abelian subgroups of index two. More applications are given in later papers.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 [Pg2] and motivated by conjugacy relations among p-subgroups of a given finite group. A saturated fusion system is reduced if it has no proper normal subsystem of p-power index, no proper normal subsystem of index prime to p, and no nontrivial normal p-subgroup. (All three of these concepts are defined by analogy with finite groups.) Reduced fusion systems need not be simple, in that they can have proper nontrivial normal subsystems. They were introduced by us in [AOV] as forming a class of fusion systems which is small enough to be manageable, but still large enough to detect any fusion systems (reduced or not) which are "exotic" (not defined via conjugacy relations in any finite group).When G is a finite group and S ∈ Syl p (G), the version of Alperin's fusion theorem shown by Goldschmidt [Gd1] says that all G-conjugacy relations among subgroups of S are generated by Aut G (S) (automorphisms induced by conjugation in G), together with Aut G (P ) for certain "essential" proper subgroups of S, and restrictions of such automorphisms. There is a version of this result for abstract fusion systems (see Theorem 1.2), which says that a fusion system F is generated by F-automorphisms of F-essential subgroups (Definition 1.1). Our goal in this and our other papers is to study, and to classify in certain cases, reduced fusion systems from the point of view of their essential subgroups and generating automorphisms.This point of view was introduced in [OV], where two of us described how fusion systems over a given 2-group S could be classified by first listing the subgroups of S which potentially could be essential, using Bender's theorem on groups with strongly embedded subgroups. When we try to extend those methods to larger classes of groups, it is useful to search for pairs of essential subgroups via theorems of Goldschmidt and Fan classifying certain types of amalgams.

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“…A fusion system is saturated if it satisfies certain additional conditions. Rather than listing those conditions here, we refer to [AKO,Definition I.2.2] or our earlier paper [AOV].…”

Section: Saturated Fusion Systemsmentioning

confidence: 99%

“…[AKO,Theorem I.2.3]). An abstract fusion system F over S is called realizable if F = F S (G) for some finite group G with S ∈ Syl p (G), and is called exotic otherwise.…”

Section: Saturated Fusion Systemsmentioning

confidence: 99%

“…We refer to [AKO,Proposition A.7] for a very brief survey of some of the properties of strongly p-embedded subgroups, and to [A1, § 46] or [Sz2,§ 6.4] for more details. Definition 1.…”

Section: Saturated Fusion Systemsmentioning

confidence: 99%

Section: Saturated Fusion Systemsmentioning

confidence: 99%

“…See, e.g., [AKO,Corollary I.7.5]. The second equivalence follows upon checking that the image of foc(F) in S/hyp(F) is precisely its commutator subgroup (cf.…”

Section: (B) [G θ] ≤ Fr(p )mentioning

confidence: 99%

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