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Permutation group
In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions 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'').〔The notations S''M'' and S''M'' are also used.〕 The term ''permutation group'' thus means a subgroup of the symmetric group. If ''M'' = then, ''Sym''(''M''), the ''symmetric group on n letters'' is usually denoted by ''Sn''.
The way in which the elements of a permutation group permute the elements of the set is called its group action. Group actions have applications in the study of symmetries, combinatorics and many other branches of mathematics, physics and chemistry.
== Basic properties and terminology ==
Being a subgroup of a symmetric group, all that is necessary for a set of permutations to satisfy the group axioms and be a permutation group is that it contain the identity permutation, the inverse permutation of each permutation it contains, and be closed under composition of its permutations. A general property of finite groups implies that a finite nonempty subset of a symmetric group is again a group if and only if it is closed under the group operation.
The degree of a group of permutations of a finite set is the 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, the order of any finite permutation group of degree ''n'' must divide ''n''! (''n''-factorial, the order of the symmetric group ''S''''n'').
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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