construct a finitely generated Abelian group
find all Abelian groups of a given order
AbelianGroup( [ t1, t2, ... ], formopt )
AbelianGroup( [ r, [ t1, t2, ... ] ], formopt )
AllAbelianGroups( n, formopt )
a non-negative integer
a positive integer
(optional) equation of the form form = F, where F is either "permgroup" or "fpgroup" (the default)
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.
Error, (in AbelianGroup) Abelian group must be finite to be represented as a permutation group
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.
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