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

Abstract

The minimal faithful permutation degree µ(G) of a finite group G is the least nonnegative integer n such that G embeds in the symmetric group Sym(n). Clearly µ(G × H) ≤ µ(G) + µ(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 × H embeds in Sym(15). Saunders (2010) produces an infinite family of examples of permutation groups G and H where µ(G × H) < µ(G) + µ(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 µ(G × H) = µ(G) + µ(H) for all groups G and H such that µ(G × H) ≤ 9.