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

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 $\Sym(n)$. 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 $\Sym(15)$. 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 $\Sym(10)$. 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 $\mu(G\times H)\le 9$.