**Cycle (mathematics)**

In mathematics, and in particular in group theory, a **cycle** is a permutation of the elements of some set *X* which maps the elements of some subset *S* of *X* to each other in a cyclic fashion, while fixing (i.e., mapping to themselves) all other elements of *X*. For example, the permutation of {1, 2, 3, 4} that sends 1 to 3, 2 to 4, 3 to 2 and 4 to 1 is a cycle, while the permutation that sends 1 to 3, 2 to 4, 3 to 1 and 4 to 2 is not (it separately permutes the pairs {1, 3} and {2, 4}). The set *S* is called the orbit of the cycle.

Read more about Cycle (mathematics): Definition, Basic Properties, Transpositions

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... One of the main results on symmetric groups states that either all of the decompositions of a given permutation into transpositions have an even number of transpositions, or they all have an odd number of transpositions, that allows to define the parity of a permutation. ...

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