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 $\sigma (s)\neq s$ is a finite subset of $S$.

For a finite set, every permutation is finitary. For an infinite set, not all permutations are 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