The Minimal Faithful Permutation Degree For A Direct Product Obeying An Inequality Condition
The minimal faithful permutation degree (G) of a finite group G is the least nonnegative integer n such that G embeds in the symmetric group Sym(n). Clearly (G x H) (G) + (H) for all finite groups G and H. In 1975, Wright () proved that equality occurs when G and H are nilpotent and exhibited an example of strict inequality where G x H embeds in Sym(15). In 2010 Saunders () produced an infinite family of examples of permutation groups G and H where (G x H) < (G) + (H), including the example of Wright's as a special case. The smallest groups in Saunders' class embed in Sym(10). In this article, we prove that 10 is minimal in the sense that (G x H)=(G) + (H) for all groups G and H such that (G x H) <= 9.