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

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 $\text{Sym}(n)$. We prove that if $H$ is a group then $\mu (G)=\mu (G\times H)$ for some group $G$ then $H$ embeds in $A\times Q^k$ for some abelian group of odd order, some generalised quaternion $2$-group and some nonnegative integer $k$. As a consequence, $\mu (G^{n+1})=\mu (G^n)$ for some nonnegative integer $n$ if and only if $G$ is trivial.