On saturated fusion systems and Brauer indecomposability of Scott modules

TL;DR

This paper investigates the conditions under which certain categories related to finite groups and modules form saturated fusion systems, focusing on Scott modules with abelian vertices and their Brauer constructions.

Contribution

It provides a sufficient condition for categories to be saturated fusion systems and proves this condition is necessary for Scott modules with abelian vertices.

Findings

A criterion for categories from Brauer pairs to be saturated fusion systems.

A characterization of when the category _P(G) is saturated.

Necessary and sufficient conditions for Scott modules with abelian vertices.

Abstract

Let $p$ be a prime number, $G$ a finite group, $P$ a $p$-subgroup of $G$ and $k$ an algebraically closed field of characteristic $p$. We study the relationship between the category $\Ff_P(G)$ and the behavior of $p$-permutation $kG$-modules with vertex $P$ under the Brauer construction. We give a sufficient condition for $\Ff_P(G)$ to be a saturated fusion system. We prove that for Scott modules with abelian vertex, our condition is also necessary. In order to obtain our results, we prove a criterion for the categories arising from the data of $(b, G)$-Brauer pairs in the sense of Alperin-Brou\'e and Brou\'e-Puig to be saturated fusion systems on the underlying $p$-group.