Realizing fusion systems inside finite groups

TL;DR

This paper demonstrates that any fusion system, regardless of saturation, can be embedded within the fusion system of a finite group, expanding the understanding of the relationship between fusion systems and finite groups.

Contribution

It extends previous work by showing all fusion systems can be realized as subcategories of finite group fusion systems, regardless of saturation.

Findings

Any fusion system can be embedded in a finite group fusion system

Extends previous results to non-saturated fusion systems

Provides a broader realization framework for fusion systems

Abstract

We show that every (not necessarily saturated) fusion system can be realized as a full subcategory of the fusion system of a finite group. This result extends our previous work \cite{Park2010} and complements the related result \cite{LearyStancu2007} by Leary and Stancu.