Let S be a set. A cycle is a permutationMathworldPlanetmath (bijective function of a set onto itself) such that there exist distinct elements a1,a2,,ak of S such that

f(ai)=ai+1  and  f(ak)=a1

that is

f(a1) = a2
f(a2) = a3
f(ak) = a1

and f(x)=x for any other element of S.

This can also be pictured as




for any other element xS, where represents the action of f.

One of the basic results on symmetric groupsMathworldPlanetmathPlanetmath says that any finite permutation can be expressed as productPlanetmathPlanetmath of disjoint cycles.

Title cycle
Canonical name Cycle1
Date of creation 2013-03-22 12:24:23
Last modified on 2013-03-22 12:24:23
Owner yark (2760)
Last modified by yark (2760)
Numerical id 10
Author yark (2760)
Entry type Definition
Classification msc 03-00
Classification msc 05A05
Classification msc 20F55
Related topic Permutation
Related topic SymmetricGroup
Related topic TranspositionMathworldPlanetmath
Related topic Group
Related topic SubgroupMathworldPlanetmathPlanetmath
Related topic DihedralGroup
Related topic CycleNotation
Related topic PermutationNotation