On permutation characters of finite group

TL;DR

This paper proves that all permutation characters of a finite group are monomial if and only if the group is solvable, resolving a question about the structure of such characters.

Contribution

It establishes a characterization of solvable groups through the monomiality of permutation characters, answering a previously posed question.

Findings

All permutation characters are monomial iff the group is solvable.

The result provides a new criterion for group solvability based on character theory.

The paper solves a specific open problem in the theory of finite groups.

Abstract

Let $G$ be a finite group and \( M \) be a maximal subgroup of \( G \). We call every irreducible constituent \( \chi \) of \( (1_M)^G \) a \( \mathcal{P} \)-character of \( G \) with respect to \( M \). In this paper, we prove that all $\mathcal{P}$-characters of $G$ are monomial if and only if $G$ is solvable, which solves a question posed by Qian and Yang.