The group of permutations of $X$ (that is the automorphism group of $X$ in $Set$) is the symmetric group (or permutation group) on $X$. This group may be denoted $S_X$, $\Sigma_X$, or $X!$. When $X$ is thefinite set$[n]$ with $n$ elements, one typically writes $S_n$ or $\Sigma_n$; note that this group has $n!$ elements.

In combinatorics, one often wants a slight generalisation. Given a natural number$r$, an $r$-permutation from $X$ is an injective function from $[r]$ to $X$, that is a list of $r$ distinct elements of $X$. Then an $n$-permutation from $[n]$ is the same as a permutation of $[n]$. (That an injective function from $X$ to itself must be invertible characterises $X$ as a Dedekind-finite set.)