Realizability and tameness of fusion systems

Abstract

A saturated fusion system over a finite (Formula presented.) -group (Formula presented.) is a category whose objects are the subgroups of (Formula presented.) and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms. A fusion system over (Formula presented.) is realized by a finite group (Formula presented.) if (Formula presented.) is a Sylow (Formula presented.) -subgroup of (Formula presented.) and morphisms in the category are those induced by conjugation in (Formula presented.). 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.