Rational and Quasi-Permutation Representations of Holomorph of Cyclic p-Groups

TL;DR

This paper investigates the minimal degrees of faithful permutation and quasi-permutation matrix representations of the holomorph of cyclic p-groups, providing explicit descriptions of irreducible representations and showing these degrees are equal to the group's order.

Contribution

It explicitly describes all irreducible representations of the holomorph of cyclic p-groups over complex and rational fields and proves the minimal degrees coincide and equal the group's order.

Findings

All irreducible representations over a9 and a9a0 fields are explicitly described.

The minimal degrees of faithful representations are equal to the order of the group.

The degrees are computed as p^n for the holomorph of cyclic groups of order p^n.

Abstract

For a finite group $G$, let $p(G)$ denote the minimal degree of a faithful permutation representation of $G$. The minimal degree of a faithful representation of $G$ by quasi-permutation matrices over the fields $\mathbb{C}$ and $\mathbb{Q}$ are denoted by $c(G)$ and $q(G)$ respectively. In general $c(G)\leq q(G)\leq p(G)$ and either inequality may be strict. In this paper, we study the representation theory of the group $G =$ Hol$(C_{p^{n}})$, which is the holomorph of a cyclic group of order $p^n$, $p$ a prime. This group is metacyclic when $p$ is odd and metabelian but not metacyclic when $p=2$ and $n \geq 3$. We explicitly describe the set of all isomorphism types of irreducible representations of $G$ over the field of complex numbers $\mathbb{C}$ as well as the isomorphism types over the field of rational numbers $\mathbb{Q}$. We compute the Wedderburn decomposition of the rational group algebra of $G$. Using the descriptions of the irreducible representations of $G$ over $\mathbb{C}$ and over $\mathbb{Q}$, we show that $c(G) = q(G) = p(G) = p^n$ for any prime $p$. The proofs are often different for the case of $p$ odd and $p=2$.