The Smallest Faithful Permutation Degree for a Direct Product obeying an Inequality Condition

David Easdown 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 \(\mathrm{Sym}(n)\). Clearly \(\mu(G \times H) \le \mu(G) + \mu(H)\) for all finite groups \(G\) and \(H\). Wright (1975) proves that equality occurs when \(G\) and \(H\) are nilpotent and exhibits an example of strict inequality where \(G\times H\) embeds in \(\mathrm{Sym}(15)\). Saunders (2010) produces an infinite family of examples of permutation groups \(G\) and \(H\) where \(\mu(G \times H) < \mu(G) + \mu(H)\), including the example of Wright's as a special case. The smallest groups in Saunders' class embed in \(\mathrm{Sym}(10)\). In this paper we prove that 10 is minimal in the sense that \(\mu(G \times H) = \mu(G) + \mu(H)\) for all groups \(G\) and \(H\) such that \(\mu(G\times H)\le 9\).

Keywords: permutation groups.

AMS Subject Classification: Primary AMS; secondary subject classification (2010): 20B35.