Realizing Finite Groups as Automizers

TL;DR

This paper demonstrates that any finite group can be realized as an automizer within a finite perfect group, utilizing fusion system techniques to construct such examples.

Contribution

It introduces a novel construction method showing all finite groups can appear as automizers in finite perfect groups using fusion system techniques.

Findings

Any finite group can be realized as an automizer in a finite perfect group.

The construction employs fusion systems and realization results from Park's work.

The method broadens understanding of automizer realizations in group theory.

Abstract

It is shown that any finite group $A$ is realizable as the automizer in a finite perfect group $G$ of an abelian subgroup whose conjugates generate $G$. The construction uses techniques from fusion systems on arbitrary finite groups, most notably certain realization results for fusion systems of the type studied originally by Park.