Symmetric group - Knowlpedia

The symmetric group $\operatorname{Sym}(X)$ of a set $X$ consists of all bijections $X\to X$ , with composition as multiplication. For $X=\{1,\ldots,n\}$ , it is denoted $S_n$ and has $n!$ elements.

Action and labeling

Evaluation gives the natural action on $X$ . A labeling of an arbitrary $n$ -element set identifies $\operatorname{Sym}(X)$ with $S_n$ ; the identification depends on the labeling.

Probability interpretation

With uniform probability on a finite set, its measure-preserving automorphism group is exactly its symmetric group. With nonuniform positive weights, only permutations preserving the weights are allowed.