A Strict Inequality for a Minimal Degree of a Direct Product

Neil Saunders

Abstract

The minimal faithful permutation degree μ(G) of a finite group G is the least non-negative integer n such that G embeds in the symmetric group Sym(n). Work of Johnson and Wright in the 1970’s established conditions for when μ(H×K) = μ(H)+μ(K), for finite groups H and K. Wright asked whether this is true for all finite groups. A counter- example of degree 15 was provided by the referee and was added as an addendum in Wright’s paper. Here we provide a counter-example of degree 12.