GroupTheory
DerivedSeries
construct the derived series of a group
LowerFittingSeries
construct the lower Fitting series of a group
DerivedLength
return the derived length of a group
FittingLength
return the Fitting length of a group
NilpotentLength
return the nilpotent length of a group
IsSoluble
determine if a group is soluble
SolubleResidual
find the soluble residual of a group
Calling Sequence
Parameters
Description
Examples
Compatibility
DerivedSeries( G )
DerivedLength( G )
LowerFittingSeries( G )
FittingLength( G )
NilpotentLength( G )
IsSoluble( G )
IsSolvable( G )
SolubleResidual( G )
SolvableResidual( G )
G

a permutation group
The derived series of a group $G$ is the descending normal series of $G$ whose terms are the successive derived subgroups, defined as follows. Let ${G}_{0}=G$ and, for $0<k$, define ${G}_{k}=\left[{G}_{k1}\,{G}_{k1}\right]$. The sequence
$G={G}_{0}\u25b9{G}_{1}\u25b9\dots \u25b9{G}_{r}$
of distinct terms is called the derived series of $G$. The number $r$ is called the derived length of $G$, and the soluble residual ${G}_{r}$ of $G$ is the last term of the derived series. If the soluble residual ${G}_{r}$ is the trivial group, then we say that $G$ is soluble (or solvable).
The DerivedSeries( G ) command constructs the derived series of a group G. The group G must be an instance of a permutation group. The derived series of G is represented by a series data structure which admits certain operations common to all series. See GroupTheory[Series].
The DerivedLength( G ) command returns the derived length of G; that is, the length of the derived series of G. This is the number of subgroup inclusions  so it is one less than the number of groups in the derived series.
The IsSoluble( G ) command (or IsSolvable( G ), as an alias) returns true if the group G is soluble, and returns the value false otherwise.
The SolubleResidual( G ) command (or SolvableResidual( G ), as an alias) returns the soluble residual ${G}_{r}$ of $G$. It can also be applied to a derived series object.
The lower Fitting series of a group $G$ is the descending normal series of $G$ whose terms are the successive nilpotent residuals. The sequence
of distinct terms is called the lower Fitting series of $G$ if ${G}_{i+1}$ is defined to be the nilpotent residual of ${G}_{i}$. Then $G$ is soluble if, and only if, this series reaches the trivial subgroup. Its length is called the Fitting length (also known as the nilpotent length) of $G$.
Note that the Fitting length of $G$ is equal to $1$ precisely when $G$ is nilpotent as, in this case, the nilpotent residual of $G$ is trivial.
The LowerFittingSeries( G ) command computes the lower Fitting series of the permutation group G. Like the derived series, the lower Fitting series of G is represented by a series data structure. Again, see GroupTheory[Series].
The FittingLength( G ) returns the Fitting length of G. The NilpotentLength( G ) command is identical, and is provided as an alias.
$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$
$G\u2254\mathrm{PermutationGroup}\left(\left\{\left[\left[1\,2\right]\right]\,\left[\left[1\,2\,3\right]\,\left[4\,5\right]\right]\right\}\right)$
${G}{\u2254}\u27e8\left({1}{\,}{2}\right){\,}\left({1}{\,}{2}{\,}{3}\right)\left({4}{\,}{5}\right)\u27e9$
$\mathrm{DerivedSeries}\left(G\right)$
$\u27e8\left({1}{\,}{2}\right){\,}\left({1}{\,}{2}{\,}{3}\right)\left({4}{\,}{5}\right)\u27e9{\u25b9}\u27e8\left({1}{\,}{3}{\,}{2}\right)\u27e9{\u25b9}\u27e8\u27e9$
$\mathrm{DerivedLength}\left(G\right)$
${2}$
$\mathrm{IsSoluble}\left(G\right)$
${\mathrm{true}}$
$\mathrm{ds}\u2254\mathrm{DerivedSeries}\left(\mathrm{AlternatingGroup}\left(4\right)\right)$
${\mathrm{ds}}{\u2254}{{\mathbf{A}}}_{{4}}{\u25b9}\u27e8\left({1}{\,}{2}\right)\left({3}{\,}{4}\right){\,}\left({1}{\,}{4}\right)\left({2}{\,}{3}\right)\u27e9{\u25b9}\u27e8\u27e9$
$\mathbf{for}\phantom{\rule[0.0ex]{0.3em}{0.0ex}}g\phantom{\rule[0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[0.0ex]{0.3em}{0.0ex}}\mathrm{ds}\phantom{\rule[0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[0.0ex]{0.0em}{0.0ex}}\phantom{\rule[0.0ex]{2.0em}{0.0ex}}\mathrm{print}\left(g\right)\phantom{\rule[0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[0.0ex]{0.3em}{0.0ex}}\mathbf{do}\:$
${{\mathbf{A}}}_{{4}}$
$\u27e8\left({1}{\,}{2}\right)\left({3}{\,}{4}\right){\,}\left({1}{\,}{4}\right)\left({2}{\,}{3}\right)\u27e9$
$\u27e8\u27e9$
$\mathrm{SolubleResidual}\left(\mathrm{ds}\right)$
$\mathrm{LowerFittingSeries}\left(\mathrm{Alt}\left(4\right)\right)$
${{\mathbf{A}}}_{{4}}{\u25b9}\u27e8\left({1}{\,}{2}\right)\left({3}{\,}{4}\right){\,}\left({1}{\,}{4}\right)\left({2}{\,}{3}\right)\u27e9{\u25b9}\u27e8\u27e9$
$\mathrm{FittingLength}\left(\mathrm{Symm}\left(4\right)\right)$
${3}$
$\mathrm{FittingLength}\left(\mathrm{DihedralGroup}\left(8\right)\right)$
${1}$
The GroupTheory[DerivedSeries], GroupTheory[DerivedLength], GroupTheory[IsSoluble] and GroupTheory[SolubleResidual] commands were introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
The GroupTheory[LowerFittingSeries], GroupTheory[FittingLength] and GroupTheory[NilpotentLength] commands were introduced in Maple 2019.
For more information on Maple 2019 changes, see Updates in Maple 2019.
See Also
GroupTheory[AlternatingGroup]
GroupTheory[DerivedSubgroup]
GroupTheory[LowerCentralSeries]
GroupTheory[NilpotentResidual]
GroupTheory[PermutationGroup]
GroupTheory[Series]
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