AbelianGroup - Maple Help

GroupTheory

 Abelian
 construct a finitely generated Abelian group
 AllAbelianGroups
 find all Abelian groups of a given order

 Calling Sequence AbelianGroup( [ t1, t2, ... ], formopt ) AbelianGroup( [ r, [ t1, t2, ... ] ], formopt ) AllAbelianGroups( n, formopt )

Parameters

 r - a non-negative integer ti - a positive integer n - a positive integer formopt - (optional) equation of the form form = F, where F is either "permgroup" or "fpgroup" (the default)

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{AbelianGroup}\left(\left[3,3\right]\right)$
 ${G}{≔}⟨{}{\mathrm{_a1}}{,}{\mathrm{_a2}}{}{\mid }{}{{\mathrm{_a1}}}^{{3}}{,}{{\mathrm{_a2}}}^{{3}}{,}{{\mathrm{_a2}}}^{{-1}}{}{{\mathrm{_a1}}}^{{-1}}{}{\mathrm{_a2}}{}{\mathrm{_a1}}{}⟩$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${9}$ (2)
 > $\mathrm{IsAbelian}\left(G\right)$
 ${\mathrm{true}}$ (3)
 > $G≔\mathrm{AbelianGroup}\left(\left[3,3\right],'\mathrm{form}'="permgroup"\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}\right){,}\left({4}{,}{5}{,}{6}\right)⟩$ (4)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${9}$ (5)
 > $\mathrm{IsAbelian}\left(G\right)$
 ${\mathrm{true}}$ (6)
 > $G≔\mathrm{AbelianGroup}\left(\left[2,\left[3,4\right]\right]\right)$
 ${G}{≔}⟨{}{\mathrm{_a1}}{,}{\mathrm{_a2}}{,}{\mathrm{_a3}}{}{\mid }{}{{\mathrm{_a2}}}^{{-1}}{}{{\mathrm{_a1}}}^{{-1}}{}{\mathrm{_a2}}{}{\mathrm{_a1}}{,}{{\mathrm{_a3}}}^{{-1}}{}{{\mathrm{_a1}}}^{{-1}}{}{\mathrm{_a3}}{}{\mathrm{_a1}}{,}{{\mathrm{_a3}}}^{{-1}}{}{{\mathrm{_a2}}}^{{-1}}{}{\mathrm{_a3}}{}{\mathrm{_a2}}{,}{{\mathrm{_a1}}}^{{12}}{}⟩$ (7)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${\mathrm{\infty }}$ (8)
 > $\mathrm{IsAbelian}\left(G\right)$
 ${\mathrm{true}}$ (9)
 > $G≔\mathrm{AbelianGroup}\left(\left[2,\left[3,4\right]\right],'\mathrm{form}'="permgroup"\right)$
 > $L≔\mathrm{AllAbelianGroups}\left(100\right)$
 ${L}{≔}⟨{}{\mathrm{_a1}}{,}{\mathrm{_a2}}{}{\mid }{}{{\mathrm{_a2}}}^{{-1}}{}{{\mathrm{_a1}}}^{{-1}}{}{\mathrm{_a2}}{}{\mathrm{_a1}}{,}{{\mathrm{_a1}}}^{{10}}{,}{{\mathrm{_a2}}}^{{10}}{}⟩{,}⟨{}{\mathrm{_a1}}{,}{\mathrm{_a2}}{}{\mid }{}{{\mathrm{_a1}}}^{{2}}{,}{{\mathrm{_a2}}}^{{-1}}{}{{\mathrm{_a1}}}^{{-1}}{}{\mathrm{_a2}}{}{\mathrm{_a1}}{,}{{\mathrm{_a2}}}^{{50}}{}⟩{,}⟨{}{\mathrm{_a1}}{,}{\mathrm{_a2}}{}{\mid }{}{{\mathrm{_a2}}}^{{-1}}{}{{\mathrm{_a1}}}^{{-1}}{}{\mathrm{_a2}}{}{\mathrm{_a1}}{,}{{\mathrm{_a1}}}^{{5}}{,}{{\mathrm{_a2}}}^{{20}}{}⟩{,}⟨{}{\mathrm{_a1}}{}{\mid }{}{{\mathrm{_a1}}}^{{100}}{}⟩$ (10)
 > $\mathrm{nops}\left(\left[L\right]\right)$
 ${4}$ (11)
 > $\mathrm{NumAbelianGroups}\left(100\right)$
 ${4}$ (12)

Compatibility

 • The GroupTheory[Abelian] command was introduced in Maple 2016.