Reduced, tame and exotic fusion systems

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

doi:10.1112/plms/pdr065

Cited by 51 publications

(151 citation statements)

References 27 publications

(142 reference statements)

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“…u t Lemma 9. 17 Let be the 2-adic valuation, and let be a primitive 2 k -th root of unity for k 2. Then 0 < .1 C / < 1.…”

Section: N C 2 Mmentioning

confidence: 99%

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Blocks of Finite Groups and Their Invariants

Sambale

2014

Lecture Notes in Mathematics

155

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“…u t Lemma 9. 17 Let be the 2-adic valuation, and let be a primitive 2 k -th root of unity for k 2. Then 0 < .1 C / < 1.…”

Section: N C 2 Mmentioning

confidence: 99%

“…Moreover, we can assume that Z.F / D 1. Now we construct the reduced fusion system of F (see Definition 2.1 in [17] [17] it is easy to see that F 1 has two essential subgroups isomorphic to C 2 2 up to conjugation. That is F 1 D F E .PSL.2; 5 2 n 1 //.…”

Section: Case (3b): a 2 M D Zmentioning

confidence: 99%

Blocks of Finite Groups and Their Invariants

Sambale

2014

Lecture Notes in Mathematics

155

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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%

“…Thus x, 1 (M) is nonabelian, hence has index ≤ p. Since |S/M| = p 2 , we have x p ,1 (M) = M and x, 1 (M) has index p. Let y ∈ 1 (M)\C M (x). Then x, y = x, 1 (M) is an A 1 -group.…”

mentioning

confidence: 99%

Saturated fusion systems over $$\mathcal {A}_2$$ A 2 -groups

Liao

Zhang

2015

Math. Z.

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“…Fusion systems over dihedral and semidihedral 2-groups have been listed by several people; cf. [AOV,§ 4.1] for the reduced case. For wreathed 2-groups, this was shown in Proposition 3.1.…”

Section: (Applied With θ = Z(r)) Thus Ymentioning

confidence: 99%

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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Reduced, tame and exotic fusion systems

Cited by 51 publications

References 27 publications

Blocks of Finite Groups and Their Invariants

Blocks of Finite Groups and Their Invariants

Saturated fusion systems over $$\mathcal {A}_2$$ A 2 -groups

Fusion systems and amalgams

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