AbelianInvariants - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Group Theory : AbelianInvariants

GroupTheory

  

AbelianInvariants

  

compute the Abelian invariants of a group

  

PrimaryInvariants

  

compute the primary invariants of a group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

AbelianInvariants( G )

PrimaryInvariants( G )

Parameters

G

-

a finitely presented group or a permutation group

Description

• 

The AbelianInvariants( G ) command computes the Abelian invariants of the abelian group G. This is returned as a list of two elements; the first entry of the list is a non-negative integer indicating the torsion-free rank, and the second is a list, B, of the orders of the cyclic factors in the canonical decomposition of the torsion subgroup. If B = [ d[1], d[2], ..., d[k] ], then the entries d[i] satisfy d[i] | d[i+1], for 1 <= i < k.

• 

The PrimaryInvariants( G ) command computes the primary invariants of the abelian group G, which represents the primary decomposition of G. This is returned as a list of two elements; the first element is the torsion-free rank (which is  if  is finite), and the second is the list of orders of the cyclic direct factors of prime power order.

• 

The group G must be a finitely presented group or a permutation group. Since a permutation group is finite, the torsion-free rank will always be equal to zero.

• 

In the case that G is a finitely presented group, the invariants of the abelianization G/[G,G] of G are computed.

Examples

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

Compatibility

• 

The GroupTheory[AbelianInvariants] command was introduced in Maple 18.

• 

For more information on Maple 18 changes, see Updates in Maple 18.

• 

The GroupTheory[PrimaryInvariants] command was introduced in Maple 2022.

• 

For more information on Maple 2022 changes, see Updates in Maple 2022.

See Also

GroupTheory

GroupTheory[DihedralGroup]

GroupTheory[HeldGroup]

 


Download Help Document