A group $G$ is supersoluble if it has a normal series with cyclic quotients. That is, there is a normal series

with each subgroup $G_i$ normal in $G$, and for which each of the quotients $\frac{G_i}{G_{i+1}}$ is cyclic.

It follows that every supersoluble group is soluble but, as the examples below illustrate, the converse is not true.

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

The IsSupersolvable( G ) command is provided as an alias.

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

$\mathrm{IsSupersoluble}\left(\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{true}$

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

$\mathrm{false}$

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

$\mathrm{true}$

$G≔\mathrm{DirectProduct}\left(\mathrm{SearchSmallGroups}\left('\mathrm{supersoluble}'\,'\mathrm{order}'=10..20\,'\mathrm{form}'=''permgroup''\right)\right)$

$G≔\mathrm{\⟨a\; permutation\; group\; on\; 558\; letters\; with\; 92\; generators\⟩}$

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

$122688296217038089632058226217949593600000000$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

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