Cycle (mathematics)

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.