PreprintMinimal permutation representations of semidirect products of groupsDavid Easdown and Michael HendriksenAbstractThe 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.
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