For a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove in ZF (without the axiom of choice) several results concerning this notion, among which are the following:

(1) For all sets x such that ${\cal S}\left( x \right)$ is Dedekind infinite, $\left| {{{\cal S}_{{\rm{fin}}}}\left( x \right)} \right| < \left| {{\cal S}\left( x \right)} \right|$ and there are no finite-to-one functions from ${\cal S}\left( x \right)$ into ${{\cal S}_{{\rm{fin}}}}\left( x \right)$, where ${{\cal S}_{{\rm{fin}}}}\left( x \right)$ denotes the set of all permutations of x which move only finitely many elements.

(2) For all sets x such that ${\cal S}\left( x \right)$ is Dedekind infinite, $\left| {{\rm{seq}}\left( x \right)} \right| < \left| {{\cal S}\left( x \right)} \right|$ and there are no finite-to-one functions from ${\cal S}\left( x \right)$ into seq (x), where seq (x) denotes the set of all finite sequences of elements of x.

(3) For all infinite sets x such that there exists a permutation of x without fixed points, there are no finite-to-one functions from ${\cal S}\left( x \right)$ into x.

(4) For all sets x, $|{[x]^2}| < \left| {{\cal S}\left( x \right)} \right|$.

For a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove by the method of permutation models that the following statements are consistent with ZF:

(1) There is an infinite set x such that $|\wp \left( x \right)| < |{\cal S}\left( x \right)| < |se{q^{1 - 1}}\left( x \right)| < |seq\left( x \right)|$, where $\wp \left( x \right)$ is the power set of x, seq (x) is the set of all finite sequences of elements of x, and seq1-1 (x) is the set of all finite sequences of elements of x without repetition.

(2) There is a Dedekind infinite set x such that $|{\cal S}\left( x \right)| < |{[x]^3}|$ and such that there exists a surjection from x onto ${\cal S}\left( x \right)$.

(3) There is an infinite set x such that there is a finite-to-one function from ${\cal S}\left( x \right)$ into x.

Forster [‘Finite-to-one maps’, J. Symbolic Logic68 (2003), 1251–1253] showed, in Zermelo–Fraenkel set theory, that if there is a finite-to-one map from ${\mathcal{P}}(A)$, the set of all subsets of a set $A$, onto $A$, then $A$ must be finite. If we assume the axiom of choice (AC), the cardinalities of ${\mathcal{P}}(A)$ and the set $S(A)$ of permutations on $A$ are equal for any infinite set $A$. In the absence of AC, we cannot make any conclusion about the relationship between the two cardinalities for an arbitrary infinite set. In this paper, we give a condition that makes Forster’s theorem, with ${\mathcal{P}}(A)$ replaced by $S(A)$, provable without AC.