On classical Clifford theory

Abstract:

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