# Finitary permutation

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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 $\operatorname{FSym}(S)$.