A group $G$ is metabelian if it is an extension of an Abelian group by another Abelian group.  Equivalently, $G$ is metabelian if its derived subgroup is Abelian. Note that every abelian group is metabelian.

The IsMetabelian( G ) command attempts to determine whether the group G is metabelian.  It returns true if G is metabelian and returns false otherwise.

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

$\mathrm{IsMetabelian}\left(\mathrm{Symm}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsMetabelian}\left(\mathrm{Alt}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{IsMetabelian}\left(\mathrm{FrobeniusGroup}\left(600\,1\right)\right)$

$\mathrm{false}$

$\mathrm{IsMetabelian}\left(\mathrm{FrobeniusGroup}\left(600\,2\right)\right)$

$\mathrm{true}$

$\mathrm{IsMetabelian}\left(\mathrm{ElementaryGroup}\left(5\,5\right)\right)$

$\mathrm{true}$

$\mathrm{IsMetabelian}\left(\mathrm{QuasicyclicGroup}\left(13\right)\right)$

$\mathrm{true}$

The GroupTheory[IsMetabelian] command was introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.

The GroupTheory[IsMetabelian] command was updated in Maple 2023.