Extensions of linking systems with p-group kernel

Oliver, Bob; Ventura, Joana

doi:10.1007/s00208-007-0104-4

Cited by 38 publications

(94 citation statements)

References 15 publications

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“…Since this contradicts (9), we now conclude that K * = ∅, and hence that F is saturated and (8) holds for all P . Now that we know F is saturated, (8) implies that H contains all subgroups which are F-centric and F-radical. Also, F 0 is normal in F (Definition 7).…”

Section: Extensions Of Linking Systems and Fusion Systems 5495mentioning

confidence: 99%

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: Extensions Of Linking Systems and Fusion Systems 5495mentioning

confidence: 99%

Extensions of linking systems and fusion systems

Oliver

2010

Trans. Amer. Math. Soc.

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“…Here to be able to work with any locality we have to relax a bit his notion of L-essential subgroups. We use this Theorem to get a generalization of the Alperin Fusion Theorem for transporter systems given by Oliver and Ventura in [OV1]. We give an example at the end how this can be applied to calculate limits of functors over a transporter system and we use it with group cohomology.…”

Section: D20 20j15; 20j06mentioning

confidence: 99%

“…The following Theorem gives a generalization of the Alperin Fusion Theorem for transporter systems [OV1,Proposition 3.10].…”

Section: Transporter Systems and Localitiesmentioning

confidence: 99%

Alperin’s fusion theorem for localities

Molinier

2017

Communications in Algebra

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In these notes we give a version of the Alperin-Goldschmidt Fusion Theorem for localities. 20D20, 20J15; 20J06The notion of a locality was introduced by Chermak [Ch] to study p-local structures of finite groups and to prove the existence and uniqueness of linking systems associated to saturated fusion systems. Along the way he proved that there is a one-to-one correspondance between localities with saturated fusion system F and transporter systems associated to F . Linking systems and transporter systems were introduced by Broto, Levi and Oliver in [BLO2] and Oliver and Ventura in [OV1] respectively. They introduced these categories to study p-completed classifying spaces of finite groups and to develop a theory of classifying spaces for saturated fusion systems. Localities give a more group-like point of view on these objects which allows us to use tools from group theory. This paper gives also an example where the setup of localities helps to prove results on transporter systems.We present here a version of the Alperin-Goldschmidt Fusion Theorem for localities which generalizes the Alperin-Goldschmidt Fusion Theorem for finite groups (a nice version can be found in [St, Theorem 1]). Chermak already gave a version of Alperin's Fusion Theorem for proper localities (i.e. localities which correspond to linking systems) in [Ch, Proposition 2.17]. Here to be able to work with any locality we have to relax a bit his notion of L-essential subgroups. We use this Theorem to get a generalization of the Alperin Fusion Theorem for transporter systems given by Oliver and Ventura in [OV1]. We give an example at the end how this can be applied to calculate limits of functors over a transporter system and we use it with group cohomology.Acknowledgement. The author is eternally grateful to Andy Chermak for so many discussions and teaching him so much about partial groups and localities. The author would also like to thank the referee for his fruitful comments.

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“…We refer the reader to [OV1,§ 3] for the precise definition as well as some of the basic properties of transporter systems. However, in contrast to the situation for linking systems (see section 3.4), there can be many transporter systems associated to the same fusion system and with the same objects.…”

Section: 2mentioning

confidence: 99%

“…As one might expect, given the name, T S (G) (see Definition 3.1) is a transporter system associated to F S (G). Conversely, if T is a transporter system associated to F and Ob (T ) includes all subgroups of S, then F is realized by the finite group G = Aut T (1) and T ∼ = T S (G) [OV1,Proposition 3.11].…”

Section: 2mentioning

confidence: 99%