The Brauer indecomposability of Scott modules with wreathed $2$-group vertices

Abstract

We give a sufficient condition for the kG-Scott module with vertex P to remain indecomposable under taking the Brauer construction for any subgroupwhere k is a field of characteristic 2, and P is a wreathed 2subgroup of a finite group G. This generalizes results for the cases where P is abelian and some others. The motivation of this paper is that the Brauer indecomposability of a p-permutation bimodule (p is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method that then can possibly lift to a splendid derived equivalence.