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GroupTheory

  

AbelianInvariants

  

compute the Abelian invariants of a finitely presented group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

AbelianInvariants( G )

Parameters

G

-

a finitely presented group

Description

• 

The AbelianInvariants( G ) command computes the Abelian invariants of the finitely presented group G, which represents the canonical decomposition of the abelianization G/[G,G] of 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 group G must be a finitely presented group.

Examples

withGroupTheory&colon;

Ga&comma;b&comma;c|a·b=b·a&comma;a2&comma;b6

Ga&comma;b&comma;ca2&comma;b-1a-1ba&comma;b6

(1)

AbelianInvariantsG

1&comma;2&comma;6

(2)

GHeldGroupform=fpgroup

GHe

(3)

AbelianInvariantsG

0&comma;

(4)

AbelianInvariantsDihedralGroup8&comma;form=fpgroup

0&comma;2&comma;2

(5)

Compatibility

• 

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

• 

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

See Also

GroupTheory

GroupTheory[DihedralGroup]

GroupTheory[HeldGroup]