A finite group $G$ is said to have perfect order classes (or subsets) if the length of each of its order classes is a divisor of the order of $G$.

Apart from the trivial group, every group with perfect order classes has even order.

The IsPerfectOrderClassesGroup( G ) command attempts to determine whether the group G is a group with perfect order classes. It returns true if G has perfect order classes, and returns false otherwise.

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

$\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{Symm}\left(3\right)\right)$

$\mathrm{true}$

$\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{Symm}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{DihedralGroup}\left(7\right)\right)$

$\mathrm{false}$

$\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{DihedralGroup}\left(9\right)\right)$

$\mathrm{true}$

$\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{CyclicGroup}\left(6\right)\right)$

$\mathrm{true}$

$\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{CyclicGroup}\left(3\right)\right)$

$\mathrm{false}$

The GroupTheory[IsPerfectOrderClassesGroup] command was introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.