Reduced fusion systems over 2-groups of small order

TL;DR

This paper proves that all reduced fusion systems over small 2-groups (up to order 2^9) are realizable by finite simple groups and are tame, using classification-based criteria.

Contribution

It establishes that reduced fusion systems over small 2-groups are always realizable and tame, extending understanding of fusion systems in this size range.

Findings

All reduced fusion systems over 2-groups of order at most 2^9 are realizable.

Such fusion systems are shown to be tame.

The proof uses criteria based on Bender's classification of groups with strongly 2-embedded subgroups.

Abstract

We prove, when $S$ is a $2$-group of order at most $2^9$, that each reduced fusion system over $S$ is the fusion system of a finite simple group and is tame. It then follows that each saturated fusion system over a $2$-group of order at most $2^9$ is realizable. What is most interesting about this result is the method of proof: we show that among $2$-groups with order in this range, the ones which can be Sylow $2$-subgroups of finite simple groups are almost completely determined by criteria based on Bender's classification of groups with strongly $2$-embedded subgroups.