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GroupTheory

 AbelianInvariants
 compute the Abelian invariants of a finitely presented group

 Calling Sequence AbelianInvariants( G )

Parameters

 G - a finitely presented group

Description

 • The AbelianInvariants( G ) command computes the Abelian invariants of the finitely presented group G, which represents the canonical decomposition of the abelianization G/[G,G] of 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 group G must be a finitely presented group.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔⟨⟨a,b,c⟩|⟨a·b=b·a,{a}^{2},{b}^{6}⟩⟩$
 ${G}{≔}⟨{}{a}{,}{b}{,}{c}{}{\mid }{}{{a}}^{{2}}{,}{{b}}^{{-1}}{}{{a}}^{{-1}}{}{b}{}{a}{,}{{b}}^{{6}}{}⟩$ (1)
 > $\mathrm{AbelianInvariants}\left(G\right)$
 $\left[{1}{,}\left[{2}{,}{6}\right]\right]$ (2)
 > $G≔\mathrm{HeldGroup}\left('\mathrm{form}'="fpgroup"\right)$
 ${G}{≔}{\mathbf{He}}$ (3)
 > $\mathrm{AbelianInvariants}\left(G\right)$
 $\left[{0}{,}\left[\right]\right]$ (4)
 > $\mathrm{AbelianInvariants}\left(\mathrm{DihedralGroup}\left(8,'\mathrm{form}'="fpgroup"\right)\right)$
 $\left[{0}{,}\left[{2}{,}{2}\right]\right]$ (5)

Compatibility

 • The GroupTheory[AbelianInvariants] command was introduced in Maple 18.