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GroupTheory

  

IsPerfectOrderClassesGroup

  

attempt to determine whether a group has perfect order classes

 

Calling Sequence

Parameters

Description

Examples

Compatibility

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

withGroupTheory:

IsPerfectOrderClassesGroupSymm3

true

(1)

IsPerfectOrderClassesGroupSymm4

false

(2)

IsPerfectOrderClassesGroupDihedralGroup7

false

(3)

IsPerfectOrderClassesGroupDihedralGroup9

true

(4)

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

IsPerfectOrderClassesGroupCyclicGroup6

true

(5)

IsPerfectOrderClassesGroupCyclicGroup3

false

(6)

Compatibility

• 

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

• 

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

See Also

GroupTheory

GroupTheory[ElementOrder]

GroupTheory[GroupOrder]

GroupTheory[OrderClassProfile]