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

Abstract

\u200eFor a finite group $G$\u200e, \u200ethree of the positive integers governing its\u200e \u200erepresentation theory over $mathbb{C}$ and over $mathbb{Q}$ are\u200e \u200e$p(G),q(G),c(G)$\u200e. \u200eHere\u200e, \u200e$p(G)$ denotes the {it minimal degree} of a\u200e \u200efaithful permutation representation of $G$\u200e. \u200eAlso\u200e, \u200e$c(G)$ and $q(G)$\u200e \u200eare\u200e, \u200erespectively\u200e, \u200ethe minimal degrees of a faithful representation\u200e \u200eof $G$ by quasi-permutation matrices over the fields $mathbb{C}$\u200e \u200eand $mathbb{Q}$\u200e. \u200eWe have $c(G)leq q(G)leq p(G)$ and\u200e, \u200ein general\u200e, \u200eeither inequality may be strict\u200e. \u200eIn this paper\u200e, \u200ewe study the\u200e \u200erepresentation theory of the group $G =$ Hol$(C_{p^{n}})$\u200e, \u200ewhich is\u200e \u200ethe {it holomorph} of a cyclic group of order $p^n$\u200e, \u200e$p$ a prime\u200e. \u200eThis group is metacyclic when $p$ is odd and metabelian but not\u200e \u200emetacyclic when $p=2$ and $n geq 3$\u200e. \u200eWe explicitly describe the set\u200e \u200eof all {it isomorphism types} of irreducible representations of $G$\u200e \u200eover the field of complex numbers $mathbb{C}$ as well as the\u200e \u200eisomorphism types over the field of rational numbers $mathbb{Q}$\u200e. \u200eWe compute the {it Wedderburn decomposition} of the rational group\u200e \u200ealgebra of $G$\u200e. \u200eUsing the descriptions of the irreducible\u200e \u200erepresentations of $G$ over $mathbb{C}$ and over $mathbb{Q}$\u200e, \u200ewe\u200e \u200eshow that $c(G) = q(G) = p(G) = p^n$ for any prime $p$\u200e. \u200eThe proofs\u200e \u200eare often different for the case of $p$ odd and $p=2$\u200e.