Important: The group package has been deprecated. Use the superseding package GroupTheory instead.

The cycle type of a permutation refers to its structure.  It can be specified by either a sample permutation with the required cycle type or by a partition of the degree.  For example, the permutation $\left[\left[1\,2\right]\,\left[3\,4\right]\,\left[5\,6\,7\right]\right]$ and the partition $\left[2\,2\,3\right]$ refer to the same cycle type.

The elements with the same cycle type are conjugates under the action of Sn, where $n$ is the degree of pg and Sn the symmetric group on $\{1,...,n\}$.

If perm is used, the function returns the number of elements of pg that have the same cycle type as perm. Only the structure of perm is considered.

If part is used, the function returns the number of elements of pg that have the cycle type described by part.

The command with(group,SnConjugates) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{group}\right)\:$

$\mathrm{pg}≔\mathrm{permgroup}\left(4\,\left\{\left[\left[1\,4\right]\right]\,\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right\}\right)\:$

$\mathrm{SnConjugates}\left(\mathrm{pg}\,\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)$

$3$

$\mathrm{SnConjugates}\left(\mathrm{pg}\,\left[2\,2\right]\right)$

$3$

$\mathrm{SnConjugates}\left(\mathrm{pg}\,\left[\left[1\,2\,3\right]\right]\right)$

$0$

$\mathrm{SnConjugates}\left(\mathrm{pg}\,\left[3\right]\right)$

$0$