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

David Easdown and Neil Saunders

Abstract

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 $$\mathrm{Sym}(n)$$. 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 $$\mathrm{Sym}(15)$$. 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 $$\mathrm{Sym}(10)$$. In this paper we prove that 10 is minimal in the sense that $$\mu(G \times H) = \mu(G) + \mu(H)$$ for all groups $$G$$ and $$H$$ such that $$\mu(G\times H)\le 9$$.

Keywords: permutation groups.

: Primary AMS; secondary subject classification (2010): 20B35.

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 Thursday, October 30, 2014