Clifford theory and endoisomorphisms

Abstract

This paper describes the relation between Clifford theory of finite groups over any field, and the Brauer–Clifford group. Let G and G¯ be finite groups, let π:G→G¯ be a surjective group homomorphism with kernel H, and let F be any field. Let K be any extension field of F and let S be an irreducible KH-module. We show that to S is associated in a natural way a specific element [[S,π,F]] of a Brauer–Clifford group defined over π and F. As a tool to prove the existence of this association, and to study its properties, we use endoisomorphisms. These are simply certain isomorphisms of related endomorphism algebras as G¯-algebras over F. We show that two modules for two different finite groups have the same (in an appropriate sense) element of the Brauer–Clifford group if and only if there exists an appropriate endoisomorphism.