Lifting Brauer indecomposability of a Scott module

TL;DR

This paper proves that under certain conditions involving a normal subgroup and solvability, the Brauer indecomposability of Scott modules lifts from a subgroup to the whole group.

Contribution

It establishes a new lifting property for Brauer indecomposability of Scott modules in groups with a normal subgroup of $p'$-index and solvable quotient.

Findings

Brauer indecomposability lifts from subgroup to group under specified conditions

Provides applications in modular representation theory

Extends understanding of Scott modules in group theory

Abstract

It is proven that if a finite group $G$ has a normal subgroup $H$ with $p'$-index (where $p$ is a prime) and $G/H$ is solvable, then for a $p$-subgroup $P$ of $H$, if the Scott $kH$-module with vertex $P$ is Brauer indecomposable, then so is the Scott $kG$-module with vertex $P$, where $k$ is a field of characteristic $p>0$. This has several applications.