Editing Permutation group (section)

{{short description|Group whose operation is composition of permutations}}
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In [[mathematics]], a '''permutation group''' is a [[group (mathematics)|group]] ''G'' whose elements are [[permutation]]s of a given [[Set (mathematics)|set]] ''M'' and whose [[group operation]] is the composition of permutations in ''G'' (which are thought of as [[bijective function]]s from the set ''M'' to itself). The group of ''all'' permutations of a set ''M'' is the [[symmetric group]] of ''M'', often written as Sym(''M'').<ref>The notations '''S'''<sub>''M''</sub> and '''S'''<sup>''M''</sup> are also used.</ref> The term ''permutation group'' thus means a [[subgroup]] of the symmetric group. If {{nowrap|1=''M'' = {1, 2, ..., ''n''<nowiki>}</nowiki> }} then Sym(''M'') is usually denoted by S<sub>''n''</sub>, and may be called the ''symmetric group on n letters''.

By [[Cayley's theorem]], every group is [[isomorphic]] to some permutation group.

The way in which the elements of a permutation group permute the elements of the set is called its [[Group action (mathematics)|group action]]. Group actions have applications in the study of [[Symmetry|symmetries]], [[combinatorics]] and many other branches of [[mathematics]], [[physics]] and chemistry.

[[Image:Rubik's cube.svg|thumb|The popular puzzle [[Rubik's cube]] invented in 1974 by [[Ernő Rubik]] has been used as an illustration of permutation groups.  Each rotation of a layer of the cube results in a [[permutation]] of the surface colors and is a member of the group.  The permutation group of the cube is called the [[Rubik's cube group]].]]