compute the Abelian invariants of a group
compute the primary invariants of a group
AbelianInvariants( G )
PrimaryInvariants( G )
a finitely presented group or a permutation group
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, d, ..., 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 0 if G 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.
G ≔ a,b,c|`.`⁡a,b=`.`⁡b,a,a2,b6
G ≔ HeldGroup⁡'form'=fpgroup
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.
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