Subgroup families controlling plocal finite groups
Abstract
A plocal finite group consists of a finite pgroup 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 psubgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as pcompleted 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 F over a finite pgroup 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 plocal 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 Fcentric Fradical subgroups (at a minimum) to the set of Fquasicentric subgroups (at a maximum). Finally, we look at constrained fusion systems, analogous to pconstrained finite groups, and prove that they in fact all arise from groups.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2004
 arXiv:
 arXiv:math/0403042
 Bibcode:
 2004math......3042B
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Group Theory;
 55R35 (Primary) 55R40;
 20D20 (Secondary)
 EPrint:
 32 pages. To appear in Proc. London Math Soc