Exercise 27, § 5

Let $Y$ be a subset of a set $X$ and let $A$ be the fixer of $Y$ in the permutation group $\mathfrak{S}_X$ of $X$. Let $M$ be the submonoid of $\mathfrak{S}_X$ consisting of the elements $s$ such that $sAs^{-1} \subset A$. Show that $M$ is a subgroup of $\mathfrak{S}_X$ if and only if one of the sets $Y$ and $X - Y$ is finite.