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 $\text{Sym}(n)$. 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 $\text{Sym}(15)$. 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 $\text{Sym}(10)$. In this paper we prove that 10 is minimal in the sense that $\mu(G \times H) = \mu(G) + \mu(H)$forallgroups$G$and$H$suchthat$\mu(G\times H)\le 9$.