 IsSupersoluble - Maple Help

GroupTheory

 IsSupersoluble
 attempt to determine whether a group is supersoluble Calling Sequence IsSupersoluble( G ) IsSupersolvable( G ) Parameters

 G - a finite group Description

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

$G={G}_{0}▹{G}_{1}▹\dots ▹{G}_{r}=1$

 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. Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{IsSupersoluble}\left(\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{true}}$ (1)

The alternating group of degree $4$ is soluble, but is not supersoluble.

 > $\mathrm{IsSupersoluble}\left(\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{IsSoluble}\left(\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{true}}$ (3)

Direct products of supersoluble groups are supersoluble.

 > $G≔\mathrm{DirectProduct}\left(\mathrm{SearchSmallGroups}\left('\mathrm{supersoluble}','\mathrm{order}'=10..20,'\mathrm{form}'="permgroup"\right)\right)$
 ${G}{≔}{\mathrm{< a permutation group on 558 letters with 92 generators >}}$ (4)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${122688296217038089632058226217949593600000000}$ (5)
 > $\mathrm{IsSupersoluble}\left(G\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{IsNilpotent}\left(G\right)$
 ${\mathrm{false}}$ (7) Compatibility

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