Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T00:54:00.273Z Has data issue: false hasContentIssue false

Subgroup families controlling p-local finite groups

Published online by Cambridge University Press:  23 August 2005

Carles Broto
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain. E-mail: [email protected], [email protected]
Natàlia Castellana
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain. E-mail: [email protected], [email protected]
Jesper Grodal
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA. E-mail: [email protected]
Ran Levi
Affiliation:
Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, United Kingdom. E-mail: [email protected]
Bob Oliver
Affiliation:
LAGA, Institut Galilée, Avenue J.-B. Clément, 93430 Villetaneuse, France. E-mail: [email protected]
Get access

Abstract

A $p$-local finite group consists of a finite $p$-group $S$, together with a pair of categories which encode ‘conjugacy’ relations among subgroups of $S$, and which are modelled on the fusion in a Sylow $p$-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as $p$-completed classifying spaces of finite groups. In this paper, we examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system $\mathcal{F}$ over a finite $p$-group $S$ is saturated can be determined by just looking at smaller classes of subgroups of $S$. We also prove that the homotopy type of the classifying space of a given $p$-local finite group is independent of the family of subgroups used to define it, in the sense that it remains unchanged when that family ranges from the set of $\mathcal{F}$-centric $\mathcal{F}$-radical subgroups (at a minimum) to the set of $\mathcal{F}$-quasicentric subgroups (at a maximum). Finally, we look at constrained fusion systems, analogous to $p$-constrained finite groups, and prove that they in fact all arise from groups.

Type
Research Article
Copyright
2005 London Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

C. Broto and N. Castellana were partially supported by MCYT grant BFM2001–2035. J. Grodal was partially supported by NSF grant DMS-0104318. R. Levi was partially supported by EPSRC grant GR/M7831. B. Oliver was partially supported by UMR 7539 of the CNRS.