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Concept# Permutation group

Summary

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.
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! since n-factorial is the order of the symmetric group Sn.
Permutation#Notations
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 is a permutation of the set then,
For instance, a particular permutation of the set {1, 2, 3, 4, 5} can be written as
this means that σ satisfies σ(1) = 2, σ(2) = 5, σ(3) = 4, σ(4) = 3, and σ(5) = 1.

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Symmetric group

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are ( factorial) such permutation operations, the order (number of elements) of the symmetric group is .

Group theory

In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

Permutation

In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set {1, 2, 3}, namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1).

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