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

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{PermutationGroup}\left(\left[\mathrm{Perm}\left(\left[\left[1,2,3,4,5,6,7,8\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,7\right],\left[2,6\right],\left[3,5\right]\right]\right)\right]\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right){,}\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right)⟩$ (1)
 > $\mathrm{LowerCentralSeries}\left(G\right)$
 $⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right){,}\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right)⟩{▹}\left[⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right){,}\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right)⟩{,}⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right){,}\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right)⟩\right]{▹}\left[⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right){,}\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right)⟩{,}\left[⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right){,}\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right)⟩{,}⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right){,}\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right)⟩\right]\right]{▹}\left[⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right){,}\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right)⟩{,}\left[⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right){,}\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right)⟩{,}\left[⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right){,}\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right)⟩{,}⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right){,}\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right)⟩\right]\right]\right]$ (2)
 > $\mathrm{IsNilpotent}\left(G\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{NilpotencyClass}\left(G\right)$
 ${3}$ (4)
 > $\mathrm{lcs}≔\mathrm{LowerCentralSeries}\left(\mathrm{AlternatingGroup}\left(4\right)\right)$
 ${\mathrm{lcs}}{≔}{{\mathbf{A}}}_{{4}}{▹}\left[{{\mathbf{A}}}_{{4}}{,}{{\mathbf{A}}}_{{4}}\right]$ (5)
 > $\mathrm{type}\left(\mathrm{lcs},'\mathrm{NormalSeries}'\right)$
 ${\mathrm{true}}$ (6)
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}H\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{lcs}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{print}\left(H\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}:$
 ${{\mathbf{A}}}_{{4}}$
 $\left[{{\mathbf{A}}}_{{4}}{,}{{\mathbf{A}}}_{{4}}\right]$ (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)
 > $\mathrm{NilpotentResidual}\left(\mathrm{DihedralGroup}\left(12\right)\right)$
 $\left[{{\mathbf{D}}}_{{12}}{,}\left[{{\mathbf{D}}}_{{12}}{,}{{\mathbf{D}}}_{{12}}\right]\right]$ (10)
 > $\mathrm{UpperCentralSeries}\left(\mathrm{DihedralGroup}\left(16\right)\right)$
 $⟨⟩{▹}⟨\left({1}{,}{9}\right)\left({2}{,}{10}\right)\left({3}{,}{11}\right)\left({4}{,}{12}\right)\left({5}{,}{13}\right)\left({6}{,}{14}\right)\left({7}{,}{15}\right)\left({8}{,}{16}\right)⟩{▹}{\dots }{▹}⟨\left({1}{,}{9}\right)\left({2}{,}{10}\right)\left({3}{,}{11}\right)\left({4}{,}{12}\right)\left({5}{,}{13}\right)\left({6}{,}{14}\right)\left({7}{,}{15}\right)\left({8}{,}{16}\right){,}\left({1}{,}{15}{,}{13}{,}{11}{,}{9}{,}{7}{,}{5}{,}{3}\right)\left({2}{,}{16}{,}{14}{,}{12}{,}{10}{,}{8}{,}{6}{,}{4}\right){,}\left({1}{,}{7}{,}{13}{,}{3}{,}{9}{,}{15}{,}{5}{,}{11}\right)\left({2}{,}{8}{,}{14}{,}{4}{,}{10}{,}{16}{,}{6}{,}{12}\right){,}\left({1}{,}{5}{,}{9}{,}{13}\right)\left({2}{,}{6}{,}{10}{,}{14}\right)\left({3}{,}{7}{,}{11}{,}{15}\right)\left({4}{,}{8}{,}{12}{,}{16}\right)⟩{▹}{{\mathbf{D}}}_{{16}}$ (11)
 > $\mathrm{UpperCentralSeries}\left(\mathrm{DihedralGroup}\left(12\right)\right)$
 $⟨⟩{▹}⟨\left({1}{,}{7}\right)\left({2}{,}{8}\right)\left({3}{,}{9}\right)\left({4}{,}{10}\right)\left({5}{,}{11}\right)\left({6}{,}{12}\right)⟩{▹}⟨\left({1}{,}{10}{,}{7}{,}{4}\right)\left({2}{,}{11}{,}{8}{,}{5}\right)\left({3}{,}{12}{,}{9}{,}{6}\right){,}\left({1}{,}{7}\right)\left({2}{,}{8}\right)\left({3}{,}{9}\right)\left({4}{,}{10}\right)\left({5}{,}{11}\right)\left({6}{,}{12}\right)⟩$ (12)
 > $\mathrm{Hypercenter}\left(\mathrm{DihedralGroup}\left(12\right)\right)$
 $⟨\left({1}{,}{10}{,}{7}{,}{4}\right)\left({2}{,}{11}{,}{8}{,}{5}\right)\left({3}{,}{12}{,}{9}{,}{6}\right){,}\left({1}{,}{7}\right)\left({2}{,}{8}\right)\left({3}{,}{9}\right)\left({4}{,}{10}\right)\left({5}{,}{11}\right)\left({6}{,}{12}\right)⟩$ (13)
 > $\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left(4{2}^{k}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}k::\mathrm{posint}$
 ${\mathrm{true}}$ (14)
 > $\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left({6}^{k}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}k::\mathrm{posint}$
 ${\mathrm{false}}$ (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.