Editing Permutation group (section)

== Basic properties and terminology ==

A ''permutation group'' is a [[subgroup]] of a [[symmetric group]]; that is, its elements are [[permutation]]s of a given set. It is thus a subset of a symmetric group that is [[closure (mathematics)|closed]] under [[function composition|composition]] of permutations, contains the [[identity permutation]], and contains the [[inverse element|inverse permutation]] of each of its elements.<ref>{{harvnb|Rotman|2006|loc=p. 148, Definition of subgroup}}</ref> A general property of finite groups implies that a finite nonempty subset of a symmetric group is a permutation group if and only if it is closed under permutation composition.<ref>{{harvnb|Rotman|2006|loc=p. 149, Proposition 2.69}}</ref>

The '''degree''' of a group of permutations of a [[finite set]] is the [[cardinality|number of elements]] in the set. The '''order''' of a group (of any type) is the number of elements (cardinality) in the group. By [[Lagrange's theorem (group theory)|Lagrange's theorem]], the order of any finite permutation group of degree ''n'' must divide ''n''! since ''n''-[[factorial]] is the order of the symmetric group ''S''<sub>''n''</sub>.