construct the commutator of two subgroups
construct the derived subgroup of a group
determine if a group is perfect
Commutator( A, B, G )
DerivedSubgroup( G )
IsPerfect( G )
a permutation group
if A and B are subgroups of a group G, then their commutator A,B is the normal subgroup of G generated by the commutators a,b, for all elements a in A and b in B.
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 G is the subgroup of G generated by the commutators a,b, as a and b range over the elements of G. Note that the derived subgroup of G is the commutator G,G. The quotient of G by its derived subgroup is called the abelianization of G, and is the largest Abelian quotient of G.
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 G 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.
A ≔ Group⁡Perm⁡1,2,3,Perm⁡1,2:
B ≔ Group⁡Perm⁡2,3,4,Perm⁡3,4:
C ≔ Commutator⁡A,B,Symm⁡4
C ≔ Commutator⁡A,B,Symm⁡5
G ≔ PermutationGroup⁡1,2,1,2,3,4,5
H ≔ DerivedSubgroup⁡AlternatingGroup⁡4
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.
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