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[1], d[2], ..., 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.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\&colon;$

$G≔⟨⟨a\&comma;b\&comma;c⟩|⟨a·b=b·a\&comma;a^2\&comma;b^6⟩⟩$

$G≔⟨a\&comma;b\&comma;c\mid a^2\&comma;b^{-1}{a}^{-1}ba\&comma;b^6⟩$

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

$\left[1\&comma;\left[2\&comma;6\right]\right]$

$\mathrm{AbelianInvariants}\left(⟨⟨a\&comma;b⟩|a^2=\left[a\&comma;b\right]⟩\right)$

$\left[1\&comma;\left[2\right]\right]$

$G≔\mathrm{HeldGroup}\left('\mathrm{form}'=''fpgroup''\right)$

$G≔\mathbf{He}$

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

$\left[0\&comma;\left[\right]\right]$

$\mathrm{AbelianInvariants}\left(\mathrm{DihedralGroup}\left(8\&comma;'\mathrm{form}'=''fpgroup''\right)\right)$

$\left[0\&comma;\left[2\&comma;2\right]\right]$

$\mathrm{AbelianInvariants}\left(\mathrm{DihedralGroup}\left(8\right)\right)$

$\left[0\&comma;\left[2\&comma;2\right]\right]$

$\mathrm{AbelianInvariants}\left(\mathrm{DicyclicGroup}\left(15\right)\right)$

$\left[0\&comma;\left[4\right]\right]$

$\mathrm{AbelianInvariants}\left(\mathrm{DicyclicGroup}\left(16\right)\right)$

$\left[0\&comma;\left[2\&comma;2\right]\right]$

$\mathrm{PrimaryInvariants}\left(\mathrm{HamiltonianGroup}\left(800\&comma;1\right)\right)$

$\left[0\&comma;\left[2\&comma;2\&comma;2\&comma;2\&comma;5\&comma;5\right]\right]$

$\mathrm{PrimaryInvariants}\left(\mathrm{AbelianGroup}\left(\left[2\&comma;\left[6\&comma;6\&comma;15\right]\right]\right)\right)$

$\left[2\&comma;\left[2\&comma;2\&comma;3\&comma;3\&comma;3\&comma;5\right]\right]$

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

For more information on Maple 18 changes, see Updates in Maple 18.

The GroupTheory[PrimaryInvariants] command was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.