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.