A group $G$ is homocyclic if it is isomorphic to a direct power of a cyclic group; that is, of the form $C_n^k$, for some positive integer $n$ and non-negative integer $k$.

The IsHomocyclic( G ) command attempts to determine whether the group G is homocyclic.  It returns true if G is homocyclic and returns false otherwise. The command may return FAIL on (most) finitely presented groups.

Cyclic groups and elementary abelian groups are homocyclic.

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

$\mathrm{IsHomocyclic}\left(\mathrm{TrivialGroup}\left(\right)\right)$

$\mathrm{true}$

$\mathrm{IsHomocyclic}\left(\mathrm{CyclicGroup}\left(15\right)\right)$

$\mathrm{true}$

$G≔\mathrm{SmallGroup}\left(36\,14\right)\:$

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

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(G\,\mathrm{DirectProduct}\left(\mathrm{CyclicGroup}\left(6\right)\,\mathrm{CyclicGroup}\left(6\right)\right)\right)$

$\mathrm{true}$

$\mathrm{IsHomocyclic}\left(\mathrm{SmallGroup}\left(4\,1\right)\right)$

$\mathrm{true}$

$\mathrm{IsHomocyclic}\left(⟨⟨a\,b⟩|⟨a^2\,b^2\,a·b=b·a⟩⟩\right)$

$\mathrm{true}$

$\mathrm{IsHomocyclic}\left(\mathrm{Symm}\left(3\right)\right)$

$\mathrm{false}$

$\mathrm{IsHomocyclic}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{false}$

The GroupTheory[IsHomocyclic] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.