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.

The AllAbelianGroups( n ) command accepts an option of the form output = "list", or output = "iterator". In the former, default case, a sequence of groups is returned. Using the output = "iterator" option causes AllAbelianGroups to return an iterator object that you can use to examine the abelian groups of order $n$ one at a time. This is useful in cases for which there is a large number of abelian groups of order $n$.

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

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

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

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

$9$

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

$\mathrm{true}$

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

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

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

$9$

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

$\mathrm{true}$

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

$G≔⟨\mathrm{\_a1}\,\mathrm{\_a2}\,\mathrm{\_a3}\,\mathrm{\_a4}\mid {\mathrm{\_a1}}^3\,{\mathrm{\_a2}}^4\,{\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}\,{\mathrm{\_a3}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a3}\mathrm{\_a1}\,{\mathrm{\_a3}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a3}\mathrm{\_a2}\,{\mathrm{\_a4}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a4}\mathrm{\_a1}\,{\mathrm{\_a4}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a4}\mathrm{\_a2}\,{\mathrm{\_a4}}^{-1}{\mathrm{\_a3}}^{-1}\mathrm{\_a4}\mathrm{\_a3}⟩$

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

$\mathrm{\infty }$

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

$\mathrm{true}$

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

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

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

$L≔⟨\mathrm{\_a1}\,\mathrm{\_a2}\,\mathrm{\_a3}\,\mathrm{\_a4}\mid {\mathrm{\_a1}}^2\,{\mathrm{\_a2}}^2\,{\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}\,{\mathrm{\_a3}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a3}\mathrm{\_a1}\,{\mathrm{\_a3}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a3}\mathrm{\_a2}\,{\mathrm{\_a4}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a4}\mathrm{\_a1}\,{\mathrm{\_a4}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a4}\mathrm{\_a2}\,{\mathrm{\_a4}}^{-1}{\mathrm{\_a3}}^{-1}\mathrm{\_a4}\mathrm{\_a3}\,{\mathrm{\_a3}}^5\,{\mathrm{\_a4}}^5⟩,⟨\mathrm{\_a1}\,\mathrm{\_a2}\,\mathrm{\_a3}\mid {\mathrm{\_a1}}^2\,{\mathrm{\_a2}}^2\,{\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}\,{\mathrm{\_a3}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a3}\mathrm{\_a1}\,{\mathrm{\_a3}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a3}\mathrm{\_a2}\,{\mathrm{\_a3}}^{25}⟩,⟨\mathrm{\_a1}\,\mathrm{\_a2}\,\mathrm{\_a3}\mid {\mathrm{\_a1}}^4\,{\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}\,{\mathrm{\_a3}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a3}\mathrm{\_a1}\,{\mathrm{\_a3}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a3}\mathrm{\_a2}\,{\mathrm{\_a2}}^5\,{\mathrm{\_a3}}^5⟩,⟨\mathrm{\_a1}\,\mathrm{\_a2}\mid {\mathrm{\_a1}}^4\,{\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}\,{\mathrm{\_a2}}^{25}⟩$

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

$4$

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

$4$

$\mathrm{it}≔\mathrm{AllAbelianGroups}\left({30}^4\,'\mathrm{form}'=''permgroup''\,'\mathrm{output}'=''iterator''\right)$

$\mathrm{it}≔\mathrm{\⟨Iterator\; for\; 125\; Abelian\; Groups\; of\; Order\; 810000\⟩}$

$$\begin{gathered} \mathbf{for}G\mathbf{in}\mathrm{it}\mathbf{do} \\ \mathbf{if}\mathrm{Exponent}\left(\mathrm{SylowSubgroup}\left(2\,G\right)\right)=4\mathbf{then} \\ \mathrm{print}\left(\mathrm{AbelianInvariants}\left(G\right)\right) \\ \mathbf{end}\mathbf{if} \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$

$\left[0\,\left[15\,30\,30\,60\right]\right]$

$\left[0\,\left[3\,30\,30\,300\right]\right]$

$\left[0\,\left[3\,6\,150\,300\right]\right]$

$\left[0\,\left[3\,6\,30\,1500\right]\right]$

$\left[0\,\left[3\,6\,6\,7500\right]\right]$

$\left[0\,\left[5\,30\,30\,180\right]\right]$

$\left[0\,\left[30\,30\,900\right]\right]$

$\left[0\,\left[6\,150\,900\right]\right]$

$\left[0\,\left[6\,30\,4500\right]\right]$

$\left[0\,\left[6\,6\,22500\right]\right]$

$\left[0\,\left[5\,10\,90\,180\right]\right]$

$\left[0\,\left[10\,90\,900\right]\right]$

$\left[0\,\left[2\,450\,900\right]\right]$

$\left[0\,\left[2\,90\,4500\right]\right]$

$\left[0\,\left[2\,18\,22500\right]\right]$

$\left[0\,\left[5\,10\,30\,540\right]\right]$

$\left[0\,\left[10\,30\,2700\right]\right]$

$\left[0\,\left[2\,150\,2700\right]\right]$

$\left[0\,\left[2\,30\,13500\right]\right]$

$\left[0\,\left[2\,6\,67500\right]\right]$

$\left[0\,\left[5\,10\,10\,1620\right]\right]$

$\left[0\,\left[10\,10\,8100\right]\right]$

$\left[0\,\left[2\,50\,8100\right]\right]$

$\left[0\,\left[2\,10\,40500\right]\right]$

$\left[0\,\left[2\,2\,202500\right]\right]$

$\left[0\,\left[15\,15\,60\,60\right]\right]$

$\left[0\,\left[3\,15\,60\,300\right]\right]$

$\left[0\,\left[3\,3\,300\,300\right]\right]$

$\left[0\,\left[3\,3\,60\,1500\right]\right]$

$\left[0\,\left[3\,3\,12\,7500\right]\right]$

$\left[0\,\left[5\,15\,60\,180\right]\right]$

$\left[0\,\left[15\,60\,900\right]\right]$

$\left[0\,\left[3\,300\,900\right]\right]$

$\left[0\,\left[3\,60\,4500\right]\right]$

$\left[0\,\left[3\,12\,22500\right]\right]$

$\left[0\,\left[5\,5\,180\,180\right]\right]$

$\left[0\,\left[5\,180\,900\right]\right]$

$\left[0\,\left[900\,900\right]\right]$

$\left[0\,\left[180\,4500\right]\right]$

$\left[0\,\left[36\,22500\right]\right]$

$\left[0\,\left[5\,5\,60\,540\right]\right]$

$\left[0\,\left[5\,60\,2700\right]\right]$

$\left[0\,\left[300\,2700\right]\right]$

$\left[0\,\left[60\,13500\right]\right]$

$\left[0\,\left[12\,67500\right]\right]$

$\left[0\,\left[5\,5\,20\,1620\right]\right]$

$\left[0\,\left[5\,20\,8100\right]\right]$

$\left[0\,\left[100\,8100\right]\right]$

$\left[0\,\left[20\,40500\right]\right]$

$\left[0\,\left[4\,202500\right]\right]$

The GroupTheory[AbelianGroup] 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.