On classical Clifford theory

Abstract

Let k k be a field, let N N be a normal subgroup of a finite group H H and let M M be a completely reducible k [ N ] k[N] -module. We give sufficient conditions for a finite dimensional (finite) group crossed product k k -algebra to be a Frobenius or symmetric k k -algebra. These results imply that k [ H ] / ( J ( k [ N ] ) k [ H ] ) k[H]/(J(k[N])k[H]) and the endomorphism k k -algebra, End k [ H ] ( M H ) {\operatorname {End} _{k[H]}}({M^H}) , of the induced module M H {M^H} are symmetric k k -algebras. We also completely describe the k [ H ] k[H] -indecomposable decomposition of M H {M^H} . It follows that the head and socle of an indecomposable component of M H {M^H} are irreducible isomorphic k [ H ] k[H] -modules.