$M$-groups and Codegrees; $M_{p}$-groups and Brauer Character Degrees

TL;DR

This work investigates how the set of codegrees of irreducible characters and the degrees of Brauer characters constrain the structure of finite groups. It proves that a group with exactly three codegrees must be an $M$-group, hence solvable and monomial, and that every $M$-group is an $M_p$-group for all primes $p$. It also shows that if every nonlinear irreducible ($p$-)Brauer character of $G$ has prime degree, then for any normal subgroup $N$ either $G'N\subseteq PN$ or $N'\subseteq P$ (where $P$ is a Sylow $p$-subgroup), implying that either $G/N$ or $N$ is an $M_p$-group. The results illuminate how codegree and Brauer-character-degree constraints influence subgroup structure and solvability, with examples illustrating the limits of converses and the necessity of certain hypotheses.

Abstract

Let $G$ be a finite group and $p$ be a prime. We prove that if $G$ has three codegrees, then $G$ is an $M$-group. We prove for some prime $p$ that if every irreducible Brauer character of $G$ is a prime, then for every normal subgroup $N$ of $G$ either $G/N$ or $N$ is an $M_p$-group.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Theorem 2.1
• proof
• proof : Proof of Theorem \ref{['theorem2']}
• Proposition 2.2
• proof