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. Cycles are often denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted.

The set S is called the orbit of the cycle. Every permutation on finitely many elements can be decomposed into cycles on disjoint orbits.

The cyclic parts of a permutation are cycles, thus the second example is composed of a 3-cycle and a 1-cycle (or fixed point) and the third is composed of two 2-cycles, and denoted (1, 3) (2, 4).

Diagram of a cyclic permutation with two fixed points; a 6-cycle and two 1-cycles.

A permutation is called a cyclic permutation if and only if it has a single nontrivial cycle (a cycle of length > 1).[1]

For example, the permutation, written in two-line notation (in two ways) and also cycle notations,

Some authors restrict the definition to only those permutations which consist of one nontrivial cycle (that is, no fixed points allowed).[2]

is a cyclic permutation under this more restrictive definition, while the preceding example is not.

The orbit of a 1-cycle is called a fixed point of the permutation, but as a permutation every 1-cycle is the identity permutation.[4] When cycle notation is used, the 1-cycles are often suppressed when no confusion will result.[5]

One of the basic results on symmetric groups is 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.[a] 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.[6]

The decomposition of a permutation into a product of transpositions is obtained for example by writing the permutation as a product of disjoint cycles, and then splitting iteratively each of the cycles of length 3 and longer into a product of a transposition and a cycle of length one less:

In fact, the symmetric group is a Coxeter group, meaning that it is generated by elements of order 2 (the adjacent transpositions), and all relations are of a certain form.

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.[8] This permits the parity of a permutation to be a well-defined concept.

This article incorporates material from cycle on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.