The Minimal Faithful Permutation Degree for a Direct Product Obeying an Inequality Condition. Issue 8 (2nd August 2016) - Easdown, David; Saunders, Neil
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 . In 1975, Wright ([10 ]) proved that equality occurs when G and H are nilpotent and exhibited an example of strict inequality where G  ×  H embeds in Sym(15). In 2010 Saunders ([7 ]) produced 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 article, 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.