  For a set $x$, let $\mathcal{S}(x)$ be the set of all permutations of $x$. We
study several aspects of this notion in $\mathsf{ZF}$. The main results are as
follows:
  (1) $\mathsf{ZF}$ proves that for all sets $x$, if $\mathcal{S}(x)$ is
Dedekind infinite, then there are no finite-to-one maps from $\mathcal{S}(x)$
into $\mathcal{S}_{\mathrm{fin}}(x)$, where $\mathcal{S}_{\mathrm{fin}}(x)$ is
the set of all permutations of $x$ which move only finitely many elements.
  (2) $\mathsf{ZF}$ proves that for all sets $x$, the cardinality of
$\mathcal{S}(x)$ is strictly greater than that of $[x]^2$.
  (3) It is consistent with $\mathsf{ZF}$ that there exists an infinite set $x$
such that the cardinality of $\mathcal{S}(x)$ is strictly less than that of
$[x]^3$.
  (4) It is consistent with $\mathsf{ZF}$ that there exists an infinite set $x$
such that there is a finite-to-one map from $\mathcal{S}(x)$ into $x$.
