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GroupTheory

  

Abelian

  

construct a finitely generated Abelian group

  

AllAbelianGroups

  

find all Abelian groups of a given order

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

AbelianGroup( [ t1, t2, ... ], formopt )

AbelianGroup( [ r, [ t1, t2, ... ] ], formopt )

AllAbelianGroups( n, formopt )

Parameters

r

-

a non-negative integer

ti

-

a positive integer

n

-

a positive integer

formopt

-

(optional) equation of the form form = F, where F is either "permgroup" or "fpgroup" (the default)

Description

• 

Every finitely generated Abelian group is isomorphic to a direct sum of a free Abelian group (which is a direct sum of finitely many infinite cyclic groups), and a direct sum of finite cyclic groups.

• 

The AbelianGroup( [ t1, t2, ... ] ) command returns a finite Abelian group isomorphic to a direct sum of cyclic groups of orders t1, t2, .... The resulting group is, by default, a finitely presented group, but a permutation group may be requested in this case.

• 

The AbelianGroup( [ r, [ t1, t2, ... ] ] ) command returns a finitely generated Abelian group isomorphic to a direct sum of a free Abelian group of rank r and a direct sum of finite cyclic groups of orders t1, t2, .... If r > 0, then a finitely presented group is returned, since the group is infinite.

• 

The AllAbelianGroups( n ) command returns an expression sequence of all the abelian groups of order n, where n is a positive integer. Since n is finite, either the 'form' = "fpgroup" or 'form' = "permgroup" options may be used.

• 

The AbelianGroup and AllAbelianGroups commands accept an option of the form form = F, where F may be either of the strings "fpgroup" (the default), or "permgroup". The form = "permgroup" option may only be used in the case that the torsion-free rank r is equal to 0.

Examples

withGroupTheory:

GAbelianGroup3,3

G_a1,_a2_a13,_a23,_a2-1_a1-1_a2_a1

(1)

GroupOrderG

9

(2)

IsAbelianG

true

(3)

GAbelianGroup3,3,form=permgroup

G1,2,3,4,5,6

(4)

GroupOrderG

9

(5)

IsAbelianG

true

(6)

GAbelianGroup2,3,4

G_a1,_a2,_a3_a2-1_a1-1_a2_a1,_a3-1_a1-1_a3_a1,_a3-1_a2-1_a3_a2,_a112

(7)

GroupOrderG

(8)

IsAbelianG

true

(9)

GAbelianGroup2,3,4,form=permgroup

Error, (in AbelianGroup) Abelian group must be finite to be represented as a permutation group

LAllAbelianGroups100

L_a1,_a2_a2-1_a1-1_a2_a1,_a110,_a210,_a1,_a2_a12,_a2-1_a1-1_a2_a1,_a250,_a1,_a2_a2-1_a1-1_a2_a1,_a15,_a220,_a1_a1100

(10)

nopsL

4

(11)

NumAbelianGroups100

4

(12)

Compatibility

• 

The GroupTheory[Abelian] command was introduced in Maple 2016.

• 

For more information on Maple 2016 changes, see Updates in Maple 2016.

• 

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

• 

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

See Also

GroupTheory

GroupTheory[CyclicGroup]

GroupTheory[GroupOrder]

GroupTheory[IsAbelian]

GroupTheory[NumAbelianGroups]

with