## Absorption of Direct Factors With Respect to the Minimal Faithful Permutation Degree of a Finite Group

### David Easdown, Michael Hendriksen 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 $$\mathrm{Sym}(n)$$. We prove that if $$H$$ is a group then $$\mu(G)=\mu(G\times H)$$ for some group $$G$$ if and only if $$H$$ embeds in the direct product of some abelian group of odd order with some power of a generalised quaternion 2-group. As a consequence, no power of a nontrivial group $$G$$ can absorb a copy of $$G$$ with respect to taking the minimal faithful permutation degree.

Keywords: permutation groups, minimal degrees, direct products.

: Primary 20B35.

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 Friday, August 19, 2016