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

TL;DR

The paper investigates a claimed implication of Klingen: if the permutation characters satisfy $1_U^G = 1_V^G$, then $1_{UN}^G = 1_{VN}^G$ for a normal subgroup $N riangleleft G$. It provides a concrete counterexample to the essential step of the original proof, showing the theorem does not hold as stated. A corrected argument is then given, establishing the averaging formula $1_{UN}^G(g) = rac{1}{|N|}\,rac{}{} ext{sum}_{n i N} 1_U^G(gn)$ for all $g eq G$, derived via a counting lemma related to orbit structure (the Fried–Guralnick–Saxl lemma) and applied to the coset action. This approach supplies a robust basis for inductive arguments involving permutation characters and their behavior under normal subgroup extensions, with potential impact on group-theoretic and arithmetic contexts that rely on such averaging identities.

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.

Theorems & Definitions (3)

• Theorem 1: Theorem 1.6 (a), page 83, klingen
• Lemma 2
• Lemma 3: Fried, Guralnick, Saxl