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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

Calling Sequence

Commutator( A, B, G )

DerivedSubgroup( G )

IsPerfect( G )

Parameters

G

-

a permutation group

A

-

a permutation group

B

-

a permutation group

Description

• 

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.

Examples

withGroupTheory:

AGroupPerm1,2,3,Perm1,2:

BGroupPerm2,3,4,Perm3,4:

CCommutatorA,B,Symm4

C1,2,3,1,2,2,3,4,3,4

(1)

GroupOrderC

12

(2)

CCommutatorA,B,Symm5

C1,2,3,1,2,2,3,4,3,4

(3)

GroupOrderC

60

(4)

GPermutationGroup1,2,1,2,3,4,5

G1,2,1,2,34,5

(5)

DerivedSubgroupG

1,2,1,2,34,5,1,2,1,2,34,5

(6)

HDerivedSubgroupAlternatingGroup4

HA4,A4

(7)

GroupOrderH

4

(8)

IsPerfectAlternatingGroup6

true

(9)

GroupOrderDerivedSubgroupAlternatingGroup6

360

(10)

IsPerfectPSL3,3

true

(11)

The special linear group SL( 2, 5 ) is an example of a non-simple finite perfect group.

IsPerfectSL2,5

true

(12)

IsSimpleSL2,5

false

(13)

Compatibility

• 

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

GroupTheory[AlternatingGroup]

GroupTheory[GroupOrder]

GroupTheory[IsPerfect]

GroupTheory[PermutationGroup]