 AbelianInvariants - Maple Help

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GroupTheory

 AbelianInvariants
 compute the Abelian invariants of a group
 PrimaryInvariants
 compute the primary invariants of a group Calling Sequence AbelianInvariants( G ) PrimaryInvariants( G ) Parameters

 G - a finitely presented group or a permutation group Description

 • The AbelianInvariants( G ) command computes the Abelian invariants of the abelian group 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 PrimaryInvariants( G ) command computes the primary invariants of the abelian group G, which represents the primary decomposition of G. This is returned as a list of two elements; the first element is the torsion-free rank (which is $0$ if $G$ is finite), and the second is the list of orders of the cyclic direct factors of prime power order.
 • The group G must be a finitely presented group or a permutation group. Since a permutation group is finite, the torsion-free rank will always be equal to zero.
 • In the case that G is a finitely presented group, the invariants of the abelianization G/[G,G] of G are computed. Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔⟨⟨a,b,c⟩|⟨\mathrm{.}\left(a,b\right)=\mathrm{.}\left(b,a\right),{a}^{2},{b}^{6}⟩⟩$
 ${G}{≔}⟨{}{a}{,}{b}{,}{c}{}{\mid }{}{{a}}^{{2}}{,}{{b}}^{{-1}}{}{{a}}^{{-1}}{}{b}{}{a}{,}{{b}}^{{6}}{}⟩$ (1)
 > $\mathrm{AbelianInvariants}\left(G\right)$
 $\left[{1}{,}\left[{2}{,}{6}\right]\right]$ (2)
 > $\mathrm{AbelianInvariants}\left(⟨⟨a,b⟩|{a}^{2}=\left[a,b\right]⟩\right)$
 $\left[{1}{,}\left[{2}\right]\right]$ (3)
 > $G≔\mathrm{HeldGroup}\left('\mathrm{form}'="fpgroup"\right)$
 ${G}{≔}{\mathbf{He}}$ (4)
 > $\mathrm{AbelianInvariants}\left(G\right)$
 $\left[{0}{,}\left[\right]\right]$ (5)
 > $\mathrm{AbelianInvariants}\left(\mathrm{DihedralGroup}\left(8,'\mathrm{form}'="fpgroup"\right)\right)$
 $\left[{0}{,}\left[{2}{,}{2}\right]\right]$ (6)
 > $\mathrm{AbelianInvariants}\left(\mathrm{DihedralGroup}\left(8\right)\right)$
 $\left[{0}{,}\left[{2}{,}{2}\right]\right]$ (7)
 > $\mathrm{AbelianInvariants}\left(\mathrm{DicyclicGroup}\left(15\right)\right)$
 $\left[{0}{,}\left[{4}\right]\right]$ (8)
 > $\mathrm{AbelianInvariants}\left(\mathrm{DicyclicGroup}\left(16\right)\right)$
 $\left[{0}{,}\left[{2}{,}{2}\right]\right]$ (9)
 > $\mathrm{PrimaryInvariants}\left(\mathrm{HamiltonianGroup}\left(800,1\right)\right)$
 $\left[{0}{,}\left[{2}{,}{2}{,}{2}{,}{2}{,}{5}{,}{5}\right]\right]$ (10)
 > $\mathrm{PrimaryInvariants}\left(\mathrm{AbelianGroup}\left(\left[2,\left[6,6,15\right]\right]\right)\right)$
 $\left[{2}{,}\left[{2}{,}{2}{,}{3}{,}{3}{,}{3}{,}{5}\right]\right]$ (11) Compatibility

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