Symmetric group

The symmetric group $S_n$ is the group of all bijections of an $n$-element set, with multiplication given by function composition. There are $n!$ such permutations, since one chooses the images successively in $n,n-1,\dots ,1$ ways.

For $n\ge 3$, composition is not commutative: $(12)(23)=(123)$ but the reversed product is $(132)$, applying the rightmost permutation first. Every permutation decomposes into disjoint cycles and into transpositions. By Cayley's theorem, every finite group embeds in some symmetric group, so permutation actions provide a universal concrete model of finite group operations.

The symmetric group, often denoted as \( S_n \), is a group that consists of all possible permutations of a finite set of \( n \) elements. The group's operation is the composition of these permutations.