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, 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).

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