Cyclic permutation

In mathematics, and in particular in group theory, a cyclic permutation (or 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 (that is, mapping to themselves) all other elements of X. If S has k elements, the cycle is called a k-cycle.A permutation is called a cyclic permutation if and only if it has a single nontrivial cycle (a cycle of length > 1).One of the basic results on symmetric groups says that any permutation can be expressed as the product of disjoint cycles (more precisely: cycles with disjoint orbits); such cycles commute with each other, and the expression of the permutation is unique up to the order of the cycles. The multiset of lengths of the cycles in this expression (the cycle type) is therefore uniquely determined by the permutation, and both the signature and the conjugacy class of the permutation in the symmetric group are determined by it.A cycle with only two elements is called a transposition. For example, the permutation π = ( 1 2 3 4 1 4 3 2 ) {displaystyle pi ={egin{pmatrix}1&2&3&4\1&4&3&2end{pmatrix}}}   that swaps 2 and 4.This article incorporates material from cycle on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.