Some Remarks on Super $M_{p}$-groups

TL;DR

This paper explores the properties of super $M_{p}$-groups, focusing on conditions that characterize such groups and showing that normal subgroups of odd order are $M_{p}$-groups, contributing to the understanding of their structure.

Contribution

It introduces the concept of super $M_{p}$-groups and establishes that normal subgroups of odd order within these groups are $M_{p}$-groups, advancing group theory knowledge.

Findings

Normal subgroups of odd order in super $M_{p}$-groups are $M_{p}$-groups.

Provides conditions for a finite group to be a super $M_{p}$-group.

Enhances understanding of the structure of super $M_{p}$-groups.

Abstract

Let $G$ be a finite group and $p$ be a prime divisor of $|G|$. An irreducible $p$-Brauer character $\varphi$ of $G$ is called super-monomial if every primitive $p$-Brauer character inducing $\varphi$ is linear. The group $G$ is said to be a super $M_{p}$-group if every irreducible $p$-Brauer character of $G$ is super-monomial. In this note, we investigate the conditions under which a finite group $G$ qualifies as a super $M_{p}$-group. We demonstrate that every normal subgroup of a super $M_{p}$-group of odd order is an $M_{p}$-group.