= Finitely supported permutation
{title2=$\operatorname{supp}(\sigma)=\{n:\sigma(n)\ne n\}\text{ finite}$}
= Finitary permutation
{synonym}
A finitely supported <permutation> fixes every element outside some finite <subset>. The finitely supported <permutations> of $\mathbb N$ are the union of the finite <symmetric groups> acting on $\{1,\ldots,k\}$, and hence form a <countable set>. All <permutations> of $\mathbb N$ form an <uncountable set>: independently swapping or fixing each pair $(2j-1,2j)$ injects the <power set> of $\mathbb N$ into this larger <set>.
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