DerivedLength - Maple Help

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 DerivedSeries( G ) DerivedLength( G ) LowerFittingSeries( G ) FittingLength( G ) NilpotentLength( G ) IsSoluble( G ) IsSolvable( G ) SolubleResidual( G ) SolvableResidual( G )

Parameters

 G - a permutation group

Description

 • 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, define ${G}_{k}=\left[{G}_{k-1},{G}_{k-1}\right]$. The sequence

$G={G}_{0}▹{G}_{1}▹\dots ▹{G}_{r}$

 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

$G={G}_{0}▹{G}_{1}▹\dots ▹{G}_{r}$

 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{PermutationGroup}\left(\left\{\left[\left[1,2\right]\right],\left[\left[1,2,3\right],\left[4,5\right]\right]\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩$ (1)
 > $\mathrm{DerivedSeries}\left(G\right)$
 $⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩{▹}⟨\left({1}{,}{3}{,}{2}\right)⟩{▹}⟨⟩$ (2)
 > $\mathrm{DerivedLength}\left(G\right)$
 ${2}$ (3)
 > $\mathrm{IsSoluble}\left(G\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{ds}≔\mathrm{DerivedSeries}\left(\mathrm{AlternatingGroup}\left(4\right)\right)$
 ${\mathrm{ds}}{≔}{{\mathbf{A}}}_{{4}}{▹}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩{▹}⟨⟩$ (5)
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}g\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{ds}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{print}\left(g\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}:$
 ${{\mathbf{A}}}_{{4}}$
 $⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩$
 $⟨⟩$ (6)
 > $\mathrm{SolubleResidual}\left(\mathrm{ds}\right)$
 $⟨⟩$ (7)
 > $\mathrm{LowerFittingSeries}\left(\mathrm{Alt}\left(4\right)\right)$
 ${{\mathbf{A}}}_{{4}}{▹}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩{▹}⟨⟩$ (8)
 > $\mathrm{FittingLength}\left(\mathrm{Symm}\left(4\right)\right)$
 ${3}$ (9)
 > $\mathrm{FittingLength}\left(\mathrm{DihedralGroup}\left(8\right)\right)$
 ${1}$ (10)

Compatibility

 • 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.