Normal subgroups and permutation characters: a correction to a proof of Klingen

Abstract

Let $G$ be a finite group. For subgroups $U$ and $V$ let $1_U^G$ and $1_V^G$ be the permutation characters for the action of $G$ on the right cosets of $U$ and $V$, respectively. Let $N$ be a normal subgroup of $G$. Norbert Klingen, in his book, shows that if $1_U^G=1_V^G$, then $1_{NU}^G=1_{NV}^G$. We give a counterexample to an argument in his proof and we give a new proof of this statement.