On permutation characters of finite group

TL;DR

This paper investigates how the monomiality of permutation-characters arising from a maximal subgroup $M$—the $\mathcal{P}$-characters—forces structural constraints on the finite group $G$. It shows that if every $\mathcal{P}$-character of $G$ is monomial, then $G$ is solvable, answering a question of Qian and Yang; it also provides a $\pi$-solvable variant linking the degrees of ${\mathcal{P}_\pi}$-characters to the existence of a normal $\pi'$-complement. The work further discusses when all irreducibles are ${\mathcal{P}_\pi}$-characters, presents counterexamples, and outlines implications for the relation between $\text{Irr}_{\mathcal{P}}(G)$ and $\text{Irr}(G)$. Overall, it strengthens the connection between permutation-character constituents and solvability, with nuanced results for pi-structures and normal complements, thereby advancing our understanding of how character-theoretic properties constrain group structure.

Abstract

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

Theorems & Definitions (4)

• Theorem A
• Theorem B
• Theorem C
• Lemma 2.1