Editing Finite group (section)

===Permutation groups===
{{Main|Permutation group}}
[[File:Symmetric group 4; Cayley graph 4,9.svg|thumb|320px|A [[Cayley graph]] of the symmetric group [[v:Symmetric group S4|S<sub>4</sub>]]]]
The '''symmetric group''' S<sub>''n''</sub> on a [[finite set]] of ''n'' symbols is the [[group (mathematics)|group]] whose elements are all the [[permutations]] of the ''n'' symbols, and whose [[group operation]] is the [[function composition|composition]] of such permutations, which are treated as [[bijection|bijective functions]] from the set of symbols to itself.<ref name=Jacobson-def>{{harvnb|Jacobson|2009|p=31}}</ref> Since there are ''n''! (''n'' [[factorial]]) possible permutations of a set of ''n'' symbols, it follows that the [[Order (group theory)|order]] (the number of elements) of the symmetric group S<sub>''n''</sub> is ''n''!.