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.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{AbelianGroup}\left(\left[3\,3\right]\right)$

$G≔⟨\mathrm{\_a1}\,\mathrm{\_a2}\mid {\mathrm{\_a1}}^3\,{\mathrm{\_a2}}^3\,{\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}⟩$

$\mathrm{GroupOrder}\left(G\right)$

$9$

$\mathrm{IsAbelian}\left(G\right)$

$\mathrm{true}$

$G≔\mathrm{AbelianGroup}\left(\left[3\,3\right]\,'\mathrm{form}'=''permgroup''\right)$

$G≔⟨\left(1\,2\,3\right)\,\left(4\,5\,6\right)⟩$

$\mathrm{GroupOrder}\left(G\right)$

$9$

$\mathrm{IsAbelian}\left(G\right)$

$\mathrm{true}$

$G≔\mathrm{AbelianGroup}\left(\left[2\,\left[3\,4\right]\right]\right)$

$G≔⟨\mathrm{\_a1}\,\mathrm{\_a2}\,\mathrm{\_a3}\mid {\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}\,{\mathrm{\_a3}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a3}\mathrm{\_a1}\,{\mathrm{\_a3}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a3}\mathrm{\_a2}\,{\mathrm{\_a1}}^{12}⟩$

$\mathrm{GroupOrder}\left(G\right)$

$\mathrm{\infty }$

$\mathrm{IsAbelian}\left(G\right)$

$\mathrm{true}$

$G≔\mathrm{AbelianGroup}\left(\left[2\,\left[3\,4\right]\right]\,'\mathrm{form}'=''permgroup''\right)$

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

$L≔\mathrm{AllAbelianGroups}\left(100\right)$

$L≔⟨\mathrm{\_a1}\,\mathrm{\_a2}\mid {\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}\,{\mathrm{\_a1}}^{10}\,{\mathrm{\_a2}}^{10}⟩,⟨\mathrm{\_a1}\,\mathrm{\_a2}\mid {\mathrm{\_a1}}^2\,{\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}\,{\mathrm{\_a2}}^{50}⟩,⟨\mathrm{\_a1}\,\mathrm{\_a2}\mid {\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}\,{\mathrm{\_a1}}^5\,{\mathrm{\_a2}}^{20}⟩,⟨\mathrm{\_a1}\mid {\mathrm{\_a1}}^{100}⟩$

$\mathrm{nops}\left(\left[L\right]\right)$

$4$

$\mathrm{NumAbelianGroups}\left(100\right)$

$4$

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.