Modules induced from a normal subgroup of prime index

Abstract

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 $V\uparrow$. We describe the structure of $U$, which is semisimple when $\mathrm{char}(\mathbb{F})\ne p$ and uniserial if $\mathrm{char}(\mathbb{F})=p$. 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 ${\mathrm{End}}_{\mathbb{F}G}(V\uparrow )$ and relate the right ideal structure of ${\mathrm{End}}_{\mathbb{F}G}(V\uparrow )$ to the submodule structure of $V\uparrow$.

15 pages. This paper from 2004 is not on the arXiv