On faithful quasi-permutation representations of $VZ$-groups and Camina $p$-groups

TL;DR

This paper investigates the minimal degrees of faithful quasi-permutation and permutation representations for specific classes of finite groups, namely VZ-groups and Camina p-groups, providing new insights into their representation theory.

Contribution

It introduces the concept of minimal degrees for quasi-permutation representations and analyzes these for VZ-groups and Camina p-groups, extending understanding of their representation properties.

Findings

Determined minimal degrees for VZ-groups.

Established bounds for Camina p-groups.

Compared quasi-permutation and permutation representation degrees.

Abstract

For a finite group $G$, we denote by $\mu(G)$ and $c(G),$ the minimal degree of faithful permutation representation of $G$ and the minimal degree of faithful representation of $G$ by quasi-permutation matrices over the complex field $\mathbb{C}$. In this paper we examine $c(G)$ for $VZ$-groups and Camina $p$-groups.