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 term permutation group thus means a subgroup of the symmetric group. If M = {1, 2, ..., n} then Sym(M) is usually denoted by Sn, 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. Group actions have applications in the study of symmetries, combinatorics and many other branches of mathematics, physics and chemistry.
Basic properties and terminology A permutation group is a subgroup of a symmetric group; that is, its elements are permutations of a given set. It is thus a subset of a symmetric group that is closed under composition of permutations, contains the identity permutation, and contains the inverse permutation of each of its elements. 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. 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! since n-factorial is the order of the symmetric group Sn.
Notation
Since permutations are bijections of a set, they can be represented by Cauchy's two-line notation. This notation lists each of the elements of M in the first row, and for each element, its image under the permutation below it in the second row. If σ {\displaystyle \sigma } is a permutation of the set M = { x 1 , x 2 , … , x n } {\displaystyle M=\{x_{1},x_{2},\ldots ,x_{n}\}} then,
σ = ( x 1 x 2 x 3 ⋯ x n σ ( x 1 ) σ ( x 2 ) σ ( x 3 ) ⋯ σ ( x n ) ) . {\displaystyle \sigma ={\begin{pmatrix}x_{1}&x_{2}&x_{3}&\cdots &x_{n}\\\sigma (x_{1})&\sigma (x_{2})&\sigma (x_{3})&\cdots &\sigma (x_{n})\end{pmatrix}}.}
For instance, a particular permutation of the set {1, 2, 3, 4, 5} can be written as
σ = ( 1 2 3 4 5 2 5 4 3 1 ) ; {\displaystyle \sigma ={\begin{pmatrix}1&2&3&4&5\\2&5&4&3&1\end{pmatrix}};}
this means that σ satisfies σ(1) = 2, σ(2) = 5, σ(3) = 4, σ(4) = 3, and σ(5) = 1. The elements of M need not appear in any special order in the first row, so the same permutation could also be written as
σ = ( 3 2 5 1 4 4 5 1 2 3 ) . {\displaystyle \sigma ={\begin{pmatrix}3&2&5&1&4\\4&5&1&2&3\end{pmatrix}}.}
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