GroupTheory
Commutator
construct the commutator of two subgroups
DerivedSubgroup
construct the derived subgroup of a group
IsPerfect
determine if a group is perfect
Calling Sequence
Parameters
Description
Examples
Compatibility
Commutator( A, B, G )
DerivedSubgroup( G )
IsPerfect( G )
G
-
a permutation group
A
B
if and are subgroups of a group , then their commutator is the normal subgroup of generated by the commutators , for all elements in and in .
The Commutator( A, B, G ) command computes the commutator of the subgroups A and B of G.
The derived subgroup (also called the commutator subgroup) of a group is the subgroup of generated by the commutators , as and range over the elements of . Note that the derived subgroup of is the commutator . The quotient of by its derived subgroup is called the abelianization of , and is the largest Abelian quotient of .
The DerivedSubgroup( G ) command constructs the derived subgroup of a group G. The group G must be an instance of a permutation group.
A group is said to be perfect if it is equal to its derived subgroup. For example, every non-Abelian simple group is perfect; however, there are perfect, but non-simple groups.
The IsPerfect( G ) command returns true if G is a perfect group, and returns false otherwise.
The special linear group SL( 2, 5 ) is an example of a non-simple finite perfect group.
The GroupTheory[DerivedSubgroup] and GroupTheory[IsPerfect] commands were introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
The GroupTheory[Commutator] command was introduced in Maple 2019.
For more information on Maple 2019 changes, see Updates in Maple 2019.
See Also
GroupTheory[AlternatingGroup]
GroupTheory[GroupOrder]
GroupTheory[PermutationGroup]
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