group(deprecated)/parity - Maple Help

group(deprecated)

 parity
 find the parity of a permutation group or a permutation

 Calling Sequence parity(pg) parity(perm) parity(part)

Parameters

 pg - permutation group perm - permutation in disjoint cycle notation part - partition

Description

 • 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 $-1$ 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.

Examples

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

 > $\mathrm{with}\left(\mathrm{group}\right):$
 > $\mathrm{pg1}≔\mathrm{permgroup}\left(6,\left\{\left[\left[1,2,3\right]\right],\left[\left[2,3,4\right]\right],\left[\left[3,4,5\right]\right],\left[\left[4,5,6\right]\right]\right\}\right):$
 > $\mathrm{pg2}≔\mathrm{permgroup}\left(6,\left\{\left[\left[1,6\right]\right],\left[\left[2,6\right]\right],\left[\left[3,6\right]\right],\left[\left[4,6\right]\right],\left[\left[5,6\right]\right]\right\}\right):$
 > $\mathrm{parity}\left(\mathrm{pg1}\right)$
 ${1}$ (1)
 > $\mathrm{parity}\left(\mathrm{pg2}\right)$
 ${-1}$ (2)
 > $\mathrm{parity}\left(\left[\left[1,2\right],\left[3,4\right]\right]\right)$
 ${1}$ (3)
 > $\mathrm{parity}\left(\left[2,2\right]\right)$
 ${1}$ (4)
 > $\mathrm{parity}\left(\left[\left[1,2\right],\left[3,4,5\right]\right]\right)$
 ${-1}$ (5)
 > $\mathrm{parity}\left(\left[2,3\right]\right)$
 ${-1}$ (6)