Minimal permutation representations of semidirect products of groups

Abstract

Abstract We provide formulae for the minimal faithful permutation degree μ ⁢ ( G ) ${\mu(G)}$ of a group G that is a semidirect product of an elementary abelian p-group by a group of prime order q not equal to p. These formulae apply to the investigation of groups G with the property that there exists a nontrivial group H such that μ ⁢ ( G × H ) = μ ⁢ ( G ) ${\mu(G\times H)=\mu(G)}$ , 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 μ ⁢ ( G × H ) = μ ⁢ ( G ) ${\mu(G\times H)=\mu(G)}$ , yet G does not decompose nontrivially as a direct product.