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.

**References**

Skiena, S. *Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica.*
Reading, MA: Addison-Wesley, p. 20, 1990.

Vardi, I. *Computational Recreations in Mathematica.* Redwood City, CA: Addison-Wesley, p. 223, 1991.

© 1996-9

1999-05-25