## Cycle (Permutation)

A Subset of a Permutation whose elements trade places with one another. A cycle decomposition of a Permutation can therefore be viewed as a Class of a Permutation Group. For example, in the Permutation Group , is a 3-cycle (, , and ) and is a 1-cycle (). Every Permutation Group on symbols can be uniquely expressed as a product of disjoint cycles. The cyclic decomposition of a Permutation can be computed in Mathematica (Wolfram Research, Champaign, IL) with the function ToCycles and the Permutation corresponding to a cyclic decomposition can be computed with FromCycles. According to Vardi (1991), the Mathematica code for ToCycles is one of the most obscure ever written.

To find the number of cycles in a Permutation Group of order , take

where is the Stirling Number of the First Kind.