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

Xiaoyou Chen, A. R. Moghaddamfar

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.

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

Abstract

Let be a finite group and be a prime divisor of . An irreducible -Brauer character of is called super-monomial if every primitive -Brauer character inducing is linear. The group is said to be a super -group if every irreducible -Brauer character of is super-monomial. In this note, we investigate the conditions under which a finite group qualifies as a super -group. We demonstrate that every normal subgroup of a super -group of odd order is an -group.
Paper Structure (2 sections, 7 theorems, 6 equations)

This paper contains 2 sections, 7 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Proofs

Key Result

Theorem 1.1

Let $G$ be a group and $p$ be a prime divisor of $|G|$. (i) If all the primitive Brauer characters are linear and every proper subgroup of $G$ is an $M_{p}$-group, then $G$ is a super $M_{p}$-group. (ii) If $G$ is a super $M$-group, then $G$ is a super $M_{p}$-group for every prime $p$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['theorem1']}
  • Lemma 2.2: G. W. Mackey
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 2 more