Realizability and tameness of fusion systems

TL;DR

This paper investigates conditions under which saturated fusion systems over finite p-groups are realizable by finite groups, establishing that realizability of components implies overall realizability and that all realizable systems are tame.

Contribution

It proves that a saturated fusion system is realizable if all its components are realizable and introduces the concept that all realizable fusion systems are tame.

Findings

Realizability of a fusion system follows from the realizability of its components.

All realizable fusion systems are shown to be tame.

Results depend on the classification of finite simple groups, with some formulations independent of it.

Abstract

A saturated fusion system over a finite $p$-group $S$ is a category whose objects are the subgroups of $S$ and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms. A fusion system over $S$ is realized by a finite group $G$ if $S$ is a Sylow $p$-subgroup of $G$ and morphisms in the category are those induced by conjugation in $G$. One recurrent question in this subject is to find criteria as to whether a given saturated fusion system is realizable or not. One main result in this paper is that a saturated fusion system is realizable if all of its components (in the sense of Aschbacher) are realizable. Another result is that all realizable fusion systems are tame: a finer condition on realizable fusion systems that involves describing automorphisms of a fusion system in terms of those of some group that realizes it. Stated in this way, these results depend on the classification of finite simple groups, but we also give more precise formulations whose proof is independent of the classification.