For a prime number $p$, a finite group $G$ is $p$-soluble if it has a subnormal series every quotient of which is either a $p$-group or has order coprime to $p$.

Every finite soluble group is $p$-soluble for every prime number $p$.

The IsPSoluble( p, G ) command returns true if the group G is $p$-soluble and returns the value false if it is not.

The group G must be an instance of a permutation group.

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

$\mathrm{IsPSoluble}\left(2\,\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{true}$

$\mathrm{IsPSoluble}\left(3\,\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{true}$

$G≔\mathrm{GL}\left(3\,3\right)\:$

$\mathrm{map}\left(\mathrm{IsPSoluble}\,\left[2\,3\,11\right]\,G\right)$

$\left[\mathrm{false}\,\mathrm{false}\,\mathrm{true}\right]$

$G≔\mathrm{FrobeniusGroup}\left(14520\,2\right)\:$

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

$\mathrm{false}$

$\mathrm{map}\left(\mathrm{IsPSoluble}\,\left[2\,3\,5\,11\right]\,\mathrm{FrobeniusGroup}\left(14520\,2\right)\right)$

$\left[\mathrm{false}\,\mathrm{false}\,\mathrm{false}\,\mathrm{true}\right]$

$\mathrm{IsPSoluble}\left(2\,\mathrm{PerfectGroup}\left(936000\,2\right)\right)$

$\mathrm{false}$

The GroupTheory[IsPSoluble] command was introduced in Maple 2024.

For more information on Maple 2024 changes, see Updates in Maple 2024.