Minimal permutation representations of semidirect products of groups

David Easdown and Michael Hendriksen

Abstract

The minimal faithful permutation degree $\mu (G)$ of a finite group $G$ is the least nonnegative integer $n$ such that $G$ embeds in the symmetric group $\mathrm{Sym}(n)$. We make observations in varying degrees of generality about $\mu (G)$ when $G$ decomposes as a semidirect product, and provide exact formulae in the case that the base group is an elementary abelian $p$-group and the extending group a cyclic group of prime order $q$ not equal to $p$. For this class, we also provide a combinatorial character$isation of group isomorphism. These results contribute to the investigation of groups $G$ with the property that there exists a nontrivial group $H$ such that $\mu (G\times H)=\mu (G)$, in particular reproducing the seminal examples of Wright (1975) and Saunders (2010). Given an arbitrarily large group $H$ that is a direct product of elementary abelian groups (with mixed primes), we construct a group $G$ such that $\mu (G\times H)=\mu (G)$, yet $G$ does not decompose nontrivially as a direct product. In the case that the exponent of $H$ is a product of distinct primes, the group $G$ is a semidirect product such that the action of $G$ on each of its Sylow $p$-subgroups, where $p$ divides the order of $H$, is irreducible. This final construction relies on properties of generalised Mersenne prime numbers.

Keywords: permutation groups, semidirect products, Mersenne numbers.

AMS Subject Classification: Primary 20B35; secondary 11A41.