Editing Group theory (section)

===Permutation groups===
The first [[Class (set theory)|class]] of groups to undergo a systematic study was [[permutation group]]s. Given any set ''X'' and a collection ''G'' of [[bijection]]s of ''X'' into itself (known as ''permutations'') that is closed under compositions and inverses, ''G'' is a group [[Group action (mathematics)|acting]] on ''X''. If ''X'' consists of ''n'' elements and ''G'' consists of ''all'' permutations, ''G'' is the [[symmetric group]] S<sub>''n''</sub>; in general, any permutation group ''G'' is a [[subgroup]] of the symmetric group of ''X''. An early construction due to [[Arthur Cayley|Cayley]] exhibited any group as a permutation group, acting on itself ({{nowrap|1=''X'' = ''G''}}) by means of the left [[regular representation]].

In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for {{nowrap|''n'' ≥ 5}}, the [[alternating group]] A<sub>''n''</sub> is [[simple group|simple]], i.e. does not admit any proper [[normal subgroup]]s. This fact plays a key role in the [[Abel–Ruffini theorem|impossibility of solving a general algebraic equation of degree {{nowrap|''n'' ≥ 5}} in radicals]].