Symmetric Group

The symmetric group on a finite set $X$ is the group whose elements are all bijective functions from $X$ to $X$ and whose group operation is that of function composition. For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of degree $n$ is the symmetric group on the set $\left\{1,2,\dots ,n\right\}$, which is often denoted as $S_n$