Important: The group package has been deprecated. Use the superseding command GroupTheory[PermParity] instead.

The function determines the parity of a permutation group, an individual permutation, or a permutation with a cycle type given by a partition.  The function returns $1$ if the parity is even, and it returns $\mathrm{-1}$ if the parity is odd. The parity of a permutation is also called the sign of a permutation.

If pg is used, the function returns the parity of pg. The parity of a permutation group is even if all of its elements are even; otherwise, it is odd.

If perm is used, the function returns the parity of perm. The permutation must be in disjoint cycle notation.

If part is used, the function returns the parity of all permutations with the cycle type described by part.

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

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

$\mathrm{pg1}≔\mathrm{permgroup}\left(6\,\left\{\left[\left[1\,2\,3\right]\right]\,\left[\left[2\,3\,4\right]\right]\,\left[\left[3\,4\,5\right]\right]\,\left[\left[4\,5\,6\right]\right]\right\}\right)\:$

$\mathrm{pg2}≔\mathrm{permgroup}\left(6\,\left\{\left[\left[1\,6\right]\right]\,\left[\left[2\,6\right]\right]\,\left[\left[3\,6\right]\right]\,\left[\left[4\,6\right]\right]\,\left[\left[5\,6\right]\right]\right\}\right)\:$

$\mathrm{parity}\left(\mathrm{pg1}\right)$

$1$

$\mathrm{parity}\left(\mathrm{pg2}\right)$

$\mathrm{-1}$

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

$1$

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

$1$

$\mathrm{parity}\left(\left[\left[1\,2\right]\,\left[3\,4\,5\right]\right]\right)$

$\mathrm{-1}$

$\mathrm{parity}\left(\left[2\,3\right]\right)$

$\mathrm{-1}$