Punctured groups for exotic fusion systems

TL;DR

This paper introduces and studies punctured groups, a class of algebraic structures related to fusion systems, demonstrating their existence, properties, and applications in understanding exotic fusion systems and their localities.

Contribution

It defines punctured groups for nonidentity subgroups, proves their existence in certain cases, and applies them to analyze exotic fusion systems and their local structures.

Findings

Punctured groups exist for certain fusion systems, including some exotic ones.

The subgroup homology decomposition on the centric collection is sharp for these systems.

Most exotic fusion systems at odd primes either have a punctured group or none due to exotic normalizers.

Abstract

The transporter systems of Oliver and Ventura and the localities of Chermak are classes of algebraic structures that model the $p$-local structures of finite groups. Other than the transporter categories and localities of finite groups, important examples include centric, quasicentric, and subcentric linking systems for saturated fusion systems. These examples are however not defined in general on the full collection of subgroups of the Sylow group. We study here punctured groups, a short name for transporter systems or localities on the collection of nonidentity subgroups of a finite $p$-group. As an application of the existence of a punctured group, we show that the subgroup homology decomposition on the centric collection is sharp for the fusion system. We also prove a Signalizer Functor Theorem for punctured groups and use it to show that the smallest Benson-Solomon exotic fusion system at the prime $2$ has a punctured group, while the others do not. As for exotic fusion systems at odd primes $p$, we survey several classes and find that in almost all cases, either the subcentric linking system is a punctured group for the system, or the system has no punctured group because the normalizer of some subgroup of order $p$ is exotic. Finally, we classify punctured groups restricting to the centric linking system for certain fusion systems on extraspecial $p$-groups of order $p^3$.