Let $G$ be a finite group and $H$ a normal subgroup of prime index $p$. Let $V$ be an irreducible ${ \mathbb F}H$-module and $U$ a quotient of the induced ${ \mathbb F}G$-module $V\kern-3pt\uparrow$. We describe the structure of $U$, which is semisimple when ${ \rm char}({ \mathbb F})\ne p$ and uniserial if ${ \rm char}({ \mathbb F})=p$. Furthermore, we describe the division rings arising as endomorphism algebras of the simple components of $U$. We use techniques from noncommutative ring theory to study ${ \rm End}_{ { \mathbb F}G}(V\kern-3pt\uparrow)$ and relate the right ideal structure of ${ \rm End}_{ { \mathbb F}G}(V\kern-3pt\uparrow)$ to the submodule structure of $V\kern-3pt\uparrow$.
Categories: math.RT