Wreath products and representations of p-local finite groups

Castellana, Natàlia; Libman, Assaf

doi:10.1016/j.aim.2009.02.011

Cited by 9 publications

(14 citation statements)

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“…The following theorem generalizes [5, Theorem 4.6] and also generalizes a related result in [6]. Recall that by definition, the set of objects in a linking system L associated to a fusion system F over S is closed under F-conjugacy and overgroups and must contain among its objects all subgroups which are F-centric and Fradical.…”

Section: Definition 8 Fix a Pair Of Saturated Fusion Systemsmentioning

confidence: 81%

“…We next prove that each P ∈ H is F-conjugate to some P ∈ H such that δ(N S (P 0 )) ∈ Syl p (Aut L (P 0 )). (6) Let P fn be the set of all S 0 -conjugacy classes [P 0 ] of subgroups P 0 ≤ S 0 which are F 0 -conjugate to P 0 and fully normalized in F 0 . (If P 0 is fully normalized in F 0 , then so is every subgroup in [P 0 ].)…”

Section: ) Denotes the Equivalence Class Of The Pair (ϕ γ) Compositmentioning

confidence: 99%

“…However, when trying to do this, one quickly discovers that a fusion system alone does not contain enough information to construct extensions, at least not in a straightforward way. This is why the results in [5], [6], and [1] are all stated in terms of linking systems. Furthermore, the extensions L 0 L of linking systems which we construct are such that the geometric realization |L 0 | has the homotopy type of a finite covering space of |L| -with covering group the group of outer automorphisms by which we extended.…”

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confidence: 99%

“…The special case shown in [5,Theorem 4.6] shows how to extend a linking system by a p-group of outer automorphisms. Other special cases were used by Castellana and Libman [6] to construct wreath products of linking systems, and by Andersen, Oliver, and Ventura [1] to construct exotic fusion and linking systems under certain hypotheses. Since all three of these constructions have very similar proofs, it should be useful to have one reference which covers all of these cases, and hopefully any others which might be needed in the future.…”

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confidence: 99%

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Extensions of linking systems and fusion systems

Oliver

2010

Trans. Amer. Math. Soc.

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Abstract. We correct two errors in the statement and proof of a theorem in an earlier paper (2007), and at the same time extend that result to a more general theorem about extensions of p-local finite groups. Other special cases of this theorem have already been shown in two later papers, so we feel it will be useful to have this more general result in the literature. This paper has two purposes: to correct some errors in the statement and proof of a theorem in the earlier paper [5], and also to prove a more general version of this theorem, describing (very roughly) how to construct extensions of fusion and linking systems by groups of outer automorphisms. Special cases of this construction have been used in at least two papers written since [5].When G is a finite group and S ∈ Syl p (G), the fusion category of G is the category F S (G) whose objects consist of all subgroups of S, and whereThis gives a means of encoding the p-local structure of G: the conjugacy relations among the p-subgroups of G. The centric linking category of G is a closely related category which (among other things) provides a link between the fusion in G and the homotopy type of its p-completed classifying space. These categories motivated the definition by Puig [10] of abstract fusion systems, and by Broto, Levi, and Oliver [3] of abstract linking systems. The main theorem in this paper (Theorem 9) describes how to construct certain types of extensions of abstract fusion and linking systems. The special case shown in [5, Theorem 4.6] shows how to extend a linking system by a p-group of outer automorphisms. Other special cases were used by Castellana and Libman [6] to construct wreath products of linking systems, and by Andersen, Oliver, and Ventura [1] to construct exotic fusion and linking systems under certain hypotheses. Since all three of these constructions have very similar proofs, it should be useful to have one reference which covers all of these cases, and hopefully any others which might be needed in the future.There was an omission in the statement of [5, Theorem 4.6], in that the group S must be assumed to act on L 0 via isotypical automorphisms (Definition 5). Without this assumption, it need not induce an action on the fusion system F 0 . The error in the proof of the theorem occurs in Step 4. In that step, a certain property of

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Section: Definition 8 Fix a Pair Of Saturated Fusion Systemsmentioning

confidence: 81%

Section: ) Denotes the Equivalence Class Of The Pair (ϕ γ) Compositmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Extensions of linking systems and fusion systems

Oliver

2010

Trans. Amer. Math. Soc.

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“…Following the notation in [14], by Theorem 2.5. Therefore we have a commutative diagram up to homotopy B( S ≀ Σ n )…”

Section: Unitary Embeddings Of P-local Compact Groupsmentioning

confidence: 99%

Unitary embeddings of finite loop spaces

Cantarero

Castellana

2016

Forum Mathematicum

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Abstract. In this paper we construct faithful representations of saturated fusion systems over discrete p-toral groups and use them to find conditions that guarantee the existence of unitary embeddings of p-local compact groups. These conditions hold for the Clark-Ewing and Aguadé-Zabrodsky p-compact groups as well as some exotic 3-local compact groups. We also show the existence of unitary embeddings of finite loop spaces. IntroductionIn the theory of compact Lie groups, the existence of a faithful unitary representation for every compact Lie group is a consequence of the Peter-Weyl theorem. This paper is concerned with the existence of analogous representations for several objects in the literature which are considered to be homotopical counterparts of compact Lie groups.In 1994, W.G. Dwyer and C.W. Wilkerson [17] introduced p-compact groups. They are loop spaces which satisfy some finiteness properties at a particular prime p. For example, if G is a compact Lie group such that its group of connected components is a finite p-group, then its p-completion G , where a bijective correspondence between connected p-compact groups and reflection data over the p-adic integers was established.Many ideas from the theory of compact Lie groups have a homotopical analogue for p-compact groups. Faithful unitary representations correspond to homotopy monomorphisms at the prime p from the classifying space BX of a pcompact group into BU (n) ∧ p for some n. A homotopy monomorphism at p is map g such that the homotopy fiber F of g ∧ p is BZ/p-null, that is, the evaluation map Map(BZ/p, F ) → F is a homotopy equivalence. For simplicity, we will call such maps BX → BU (n) In this article we deal with the same question for the combinatorial structures called p-local compact groups, which encode the p-local information of some spaces at a prime p. They were introduced in [9] to model p-completed classifying spaces of compact Lie groups, p-compact groups, as well as linear torsion groups, and they 2010 Mathematics Subject Classification. 55R35, (primary), 20D20, 20C20 (secondary).

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Topological actions of wreath products

Maksymenko

2024

European Journal of Mathematics

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Wreath products and representations of p-local finite groups

Cited by 9 publications

References 21 publications

Extensions of linking systems and fusion systems

Extensions of linking systems and fusion systems

Unitary embeddings of finite loop spaces

Topological actions of wreath products

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