IsPerfectOrderClassesGroup - Maple Help

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GroupTheory

 IsPerfectOrderClassesGroup
 attempt to determine whether a group has perfect order classes

 Calling Sequence IsPerfectOrderClassesGroup( G )

Parameters

 G - a finite group

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{true}}$ (1)
 > $\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{DihedralGroup}\left(7\right)\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{DihedralGroup}\left(9\right)\right)$
 ${\mathrm{true}}$ (4)

These next two examples demonstrate that the groups with perfect order classes are closed under neither subgroups or quotients.

 > $\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{CyclicGroup}\left(6\right)\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{CyclicGroup}\left(3\right)\right)$
 ${\mathrm{false}}$ (6)

Compatibility

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