## 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.

: Primary 20B35; secondary 11A41.

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 Monday, August 24, 2015