A permutation group is a groupof orderwhose elements are permutations of the integersThe set of all permutations is labelledand called the symmetric group. A permutation group labelledis usually a subgroup of the symmetric group.
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, (1)(2)...(n1), the inverse permutation of each permutation it contains, and be closed under composition of its permutations.
Consider the following setof permutations of the set {1,2,3,4}:

The identity,

The labels 1 and 2 are interchanged, 3 and 4 are fixed.

The labels 1 and 2 are fixed, 1 and 2 are interchanged.

This permutation interchanges 1 with 2, and 3 with 4.
forms a permutation group with each element self inverse. It is isomorphic to the Klein group.
More generally, every groupis isomorphic to a permutation group by virtue of its regular action onas a set; this is the content of Cayley's theorem.
Ifandare two permutation groups, then we say thatandare isomorphic as permutation groups if there exists a bijective mapbetweenandsuch thatwithThis is equivalent toandbeing conjugate subgroups of
A 2cycle is known as a transposition. A simple transposition inis a 2cycle of the form
Every permutation p can be written as a product of simple transpositions; furthermore, the number of simple transpositions one can write a permutation as is the number of swaps needed to bring the n1 numbers in the setback to the natural orderand if the number of transpositions in p is odd or even corresponding to the oddness of p the number of swaps is also odd or even. Composing permutations has the following intuitive rules:
The set of even transposition informs a subgroup offor each