Subgroup families controlling p-local finite groups
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 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 F-centric F-radical subgroups (at a minimum) to the set of 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.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- March 2004
- DOI:
- 10.48550/arXiv.math/0403042
- arXiv:
- arXiv:math/0403042
- Bibcode:
- 2004math......3042B
- Keywords:
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- Mathematics - Algebraic Topology;
- Mathematics - Group Theory;
- 55R35 (Primary) 55R40;
- 20D20 (Secondary)
- E-Print:
- 32 pages. To appear in Proc. London Math Soc