Absorption of Direct Factors With Respect to the Minimal Faithful Permutation Degree of a Finite Group

David Easdown, Michael Hendriksen and Neil Saunders

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 prove that if $H$ is a group then $\mu (G)=\mu (G\times H)$ for some group $G$ if and only if $H$ embeds in the direct product of some abelian group of odd order with some power of a generalised quaternion 2-group. As a consequence, no power of a nontrivial group $G$ can absorb a copy of $G$ with respect to taking the minimal faithful permutation degree.

Keywords: permutation groups, minimal degrees, direct products.

AMS Subject Classification: Primary 20B35.