Subgroup Families Controlling p‐Local Finite Groups

By C. Broto, Natàlia Castellana, Jesper Grodal, Ran Levi, B. Oliver. 2004

140 citations recorded by the source corpus.

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. 2000 Mathematics Subject Classification 20J99 (primary), 55R35, 20D20 (secondary).

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dd04d2764d85826bb26db7e2c8e9d5056d23621b

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Citations in the corpus

Extensions of p-local nite groups (2007, methodology): “We recall the precise definitions of these objects in Section 1, and refer to [BLO2] and [ 5A1 ] for motivation for their study.”
Extensions of linking systems with p-group kernel (2007, methodology): “Proof Part (a) is Alperin’s fusion theorem for saturated fusion systems, in the form shown in [5, Theorem A.10]. Part (b) is proven in [ 2, Theorem 2.2 ].”