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.