Permutation group

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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 relationship is often written as (G,M). Note that the group of all permutations of a set is the symmetric group; the term permutation group is usually restricted to mean a subgroup of the symmetric group. The symmetric group of n elements is denoted by Sn; if M is any finite or infinite set, then the group of all permutations of M is often written as Sym(M).

The application of a permutation group to the elements being permuted is called its group action; it has applications in both the study of symmetries, combinatorics and many other branches of mathematics, physics and chemistry.

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Closure properties

As a subgroup of a symmetric group, all that is necessary for a permutation group to satisfy the group axioms 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 subset of a symmetric group is again a group if and only if it is closed under the group operation.

Examples

Permutations are often written in cyclic form, e.g. during cycle index computations, so that given the set M = {1,2,3,4}, a permutation g of M with g(1) = 2, g(2) = 4, g(4) = 1 and g(3) = 3 will be written as (1,2,4)(3), or more commonly, (1,2,4) since 3 is left unchanged; if the objects are denoted by a single letter or digit, commas are also dispensed with, and we have a notation such as (1 2 4).

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