Hypercenter - Maple Help

GroupTheory

 LowerCentralSeries
 construct the lower central series of a group
 UpperCentralSeries
 construct the upper central series of a group
 IsNilpotent
 determine if a group is nilpotent
 NilpotencyClass
 find the nilpotency class of a group
 NilpotentResidual
 find the nilpotency residual of a group
 Hypercenter
 find the hypercenter of a group

 Calling Sequence LowerCentralSeries( G ) UpperCentralSeries( G ) IsNilpotent( G ) NilpotencyClass( G ) NilpotentResidual( G ) Hypercenter( G )

Parameters

 G - a permutation group

Description

 • The lower central series of a group $G$ is the descending normal series of $G$ whose terms are the successive commutator subgroups, defined as follows. Let ${G}_{0}=G$ and, for $0, define ${G}_{k}=\left[G,{G}_{k-1}\right]$. The sequence

$G={G}_{0}▹{G}_{1}▹\dots ▹{G}_{c}$

 is called the lower central series of $G$. If the nilpotent residual ${G}_{c}$ is the trivial group, then we say that $G$ is nilpotent. In this case, the number $c$ is called the nilpotency class of $G$, and the nilpotent residual ${G}_{c}$ of $G$ is the last term of the lower central series.
 • The LowerCentralSeries( G ) command constructs the lower central series of a group G. The group G must be an instance of a permutation group.
 • The IsNilpotent( G ) command determines whether a group G is nilpotent.
 • The NilpotencyClass( G ) command returns the nilpotency class of G; that is, the length of the lower central series of G.
 • The NilpotentResidual( G ) command returns the nilpotent residual of a group G.
 • The upper central series of a group $G$ is the ascending normal series of $G$ whose terms are defined, recursively, as follows. Let ${G}_{0}=1$ and, for $0, define ${G}_{k}$ to be the pre-image, in $G$, of the center of the quotient group $\frac{G}{{G}_{k-1}}$.  (Thus, ${G}_{1}$ is just the center of $G$.) The sequence

$1={G}_{0}◃{G}_{1}◃\dots ◃{G}_{c}$

 is called the upper central series of $G$.
 • The UpperCentralSeries( G ) command constructs the upper central series of a group G.
 • The group $G$ is nilpotent if, and only if, the last term ${G}_{c}$ of the upper central series is equal to $G$. In general, the final term ${G}_{c}$ is called the hypercenter of $G$.
 • The Hypercenter( G ) command returns the hypercenter of a group G.
 • The Hypercentre command is provided as an alias.
 • The group G must be an instance of a permutation group.
 • Both the lower and upper central series of G are represented by a series data structure which admits certain operations common to all series.  See GroupTheory[Series].

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{PermutationGroup}\left(\left[\mathrm{Perm}\left(\left[\left[1,2,3,4,5,6,7,8\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,7\right],\left[2,6\right],\left[3,5\right]\right]\right)\right]\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{PermutationGroup}}{}\left(\left\{{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}{,}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right\}{,}{\mathrm{degree}}{=}{8}\right)$ (1)
 > $\mathrm{LowerCentralSeries}\left(G\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{LowerCentralSeries}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (2)
 > $\mathrm{IsNilpotent}\left(G\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{NilpotencyClass}\left(G\right)$
 ${3}$ (4)
 > $\mathrm{lcs}≔\mathrm{LowerCentralSeries}\left(\mathrm{AlternatingGroup}\left(4\right)\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{LowerCentralSeries}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (5)
 > $\mathrm{type}\left(\mathrm{lcs},'\mathrm{NormalSeries}'\right)$
 ${\mathrm{true}}$ (6)
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}H\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{lcs}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{print}\left(H\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}:$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{AlternatingGroup}}{}\left({4}\right)$
 ${\mathrm{Commutator}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}{,}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}{,}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (7)
 > $\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left(8\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left(12\right)\right)$
 ${\mathrm{false}}$ (9)
 > $\mathrm{NilpotentResidual}\left(\mathrm{DihedralGroup}\left(12\right)\right)$
 ${\mathrm{Commutator}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}{,}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}{,}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (10)
 > $\mathrm{UpperCentralSeries}\left(\mathrm{DihedralGroup}\left(16\right)\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{UpperCentralSeries}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (11)
 > $\mathrm{UpperCentralSeries}\left(\mathrm{DihedralGroup}\left(12\right)\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{UpperCentralSeries}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (12)
 > $\mathrm{Hypercenter}\left(\mathrm{DihedralGroup}\left(12\right)\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{PermutationGroup}}{}\left(\left\{{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}{,}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right\}{,}{\mathrm{degree}}{=}{12}{,}{\mathrm{supergroup}}{=}{\mathrm{GroupTheory}}{:-}{\mathrm{PermutationGroup}}{}\left(\left\{{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}{,}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right\}{,}{\mathrm{degree}}{=}{12}\right)\right)$ (13)
 > $\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left(4{2}^{k}\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}k::\mathrm{posint}$
 ${\mathrm{true}}$ (14)
 > $\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left({6}^{k}\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}k::\mathrm{posint}$
 ${\mathrm{false}}$ (15)
 > $\mathrm{seq}\left(\mathrm{NilpotencyClass}\left(\mathrm{QuaternionGroup}\left(n\right)\right),n=3..10\right)$
 ${2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}{,}{9}$ (16)

Compatibility

 • The GroupTheory[LowerCentralSeries] command was introduced in Maple 17.