Finitary permutation

Definition
Let $$S$$ be a set. A permutation $$\sigma$$ on $$S$$ is termed a finitary permutation if the set of points $$s \in S$$ such that $$\sigma(s) \ne s$$ is a finite subset of $$S$$.

For a finite set, every permutation is finitary. For an infinite set, all permutations are not finitary. The finitary permutations form a proper subgroup of the symmetric group on $$S$$, termed the finitary symmetric group on $$S$$ and denoted $$\operatorname{FSym}(S)$$.

Related facts

 * Infinite group has no non-identity finitary automorphism