Rational and Quasi-Permutation Representations of Holomorphs of Cyclic $p$-Groups

Abstract

‎For a finite group $G$‎, ‎three of the positive integers governing its‎ ‎representation theory over $mathbb{C}$ and over $mathbb{Q}$ are‎ ‎$p(G),q(G),c(G)$‎. ‎Here‎, ‎$p(G)$ denotes the {it minimal degree} of a‎ ‎faithful permutation representation of $G$‎. ‎Also‎, ‎$c(G)$ and $q(G)$‎ ‎are‎, ‎respectively‎, ‎the minimal degrees of a faithful representation‎ ‎of $G$ by quasi-permutation matrices over the fields $mathbb{C}$‎ ‎and $mathbb{Q}$‎. ‎We have $c(G)leq q(G)leq p(G)$ and‎, ‎in general‎, ‎either inequality may be strict‎. ‎In this paper‎, ‎we study the‎ ‎representation theory of the group $G =$ Hol$(C_{p^{n}})$‎, ‎which is‎ ‎the {it 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 {it 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 {it 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 $p=2$‎.