International Journal of Group Theory (Sep 2022)
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 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 $p=2$.
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