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 $0<k$, define $G_k=\left[G\&comma;G_{k-1}\right]$. The sequence

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 $0<k$, 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

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].

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\&colon;$

$G≔\mathrm{PermutationGroup}\left(\left\{\left[\left[1\&comma;2\right]\right]\&comma;\left[\left[1\&comma;2\&comma;3\right]\&comma;\left[4\&comma;5\right]\right]\right\}\right)$

$\mathrm{GroupTheory}:-\mathrm{PermutationGroup}\left(\left\{\mathbf{module}\left(\right)...\mathbf{end\; module}\&comma;\mathbf{module}\left(\right)...\mathbf{end\; module}\right\}\&comma;\mathrm{degree}\&equals;5\right)$

$\mathrm{LowerCentralSeries}\left(G\right)$

$\mathrm{GroupTheory}:-\mathrm{LowerCentralSeries}\left(\mathbf{module}\left(\right)...\mathbf{end\; module}\right)$

$\mathrm{NilpotencyClass}\left(G\right)$

$1$

$\mathrm{IsNilpotent}\left(G\right)$

$\mathrm{false}$

$\mathrm{lcs}≔\mathrm{LowerCentralSeries}\left(\mathrm{AlternatingGroup}\left(4\right)\right)$

$\mathrm{GroupTheory}:-\mathrm{LowerCentralSeries}\left(\mathbf{module}\left(\right)...\mathbf{end\; module}\right)$

$$\begin{gathered} \mathbf{for}g\mathbf{in}\mathrm{lcs}\mathbf{do} \\ \mathrm{print}\left(g\right) \\ \mathbf{end}\mathbf{do}\&colon; \end{gathered}$$

$\mathrm{GroupTheory}:-\mathrm{AlternatingGroup}\left(4\right)$

$\mathrm{GroupTheory}:-\mathrm{PermutationGroup}\left(\left\{\mathbf{module}\left(\right)...\mathbf{end\; module}\&comma;\mathbf{module}\left(\right)...\mathbf{end\; module}\&comma;\mathbf{module}\left(\right)...\mathbf{end\; module}\right\}\&comma;\mathrm{degree}\&equals;4\&comma;\mathrm{supergroup}\&equals;\mathrm{GroupTheory}:-\mathrm{PermutationGroup}\left(\left\{\mathbf{module}\left(\right)...\mathbf{end\; module}\&comma;\mathbf{module}\left(\right)...\mathbf{end\; module}\right\}\&comma;\mathrm{degree}\&equals;4\right)\right)$

$\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left(8\right)\right)$

$\mathrm{true}$

$\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left(12\right)\right)$

$\mathrm{false}$

$\mathrm{NilpotentResidual}\left(\mathrm{DihedralGroup}\left(12\right)\right)$

$\mathrm{GroupTheory}:-\mathrm{PermutationGroup}\left(\left\{\mathbf{module}\left(\right)...\mathbf{end\; module}\&comma;\mathbf{module}\left(\right)...\mathbf{end\; module}\right\}\&comma;\mathrm{degree}\&equals;12\&comma;\mathrm{supergroup}\&equals;\mathrm{GroupTheory}:-\mathrm{PermutationGroup}\left(\left\{\mathbf{module}\left(\right)...\mathbf{end\; module}\&comma;\mathbf{module}\left(\right)...\mathbf{end\; module}\right\}\&comma;\mathrm{degree}\&equals;12\&comma;\mathrm{supergroup}\&equals;\mathrm{GroupTheory}:-\mathrm{PermutationGroup}\left(\left\{\mathbf{module}\left(\right)...\mathbf{end\; module}\&comma;\mathbf{module}\left(\right)...\mathbf{end\; module}\right\}\&comma;\mathrm{degree}\&equals;12\right)\right)\right)$

$\mathrm{UpperCentralSeries}\left(\mathrm{DihedralGroup}\left(16\right)\right)$

$\mathrm{GroupTheory}:-\mathrm{UpperCentralSeries}\left(\mathbf{module}\left(\right)...\mathbf{end\; module}\right)$

$\mathrm{UpperCentralSeries}\left(\mathrm{DihedralGroup}\left(12\right)\right)$

$\mathrm{GroupTheory}:-\mathrm{UpperCentralSeries}\left(\mathbf{module}\left(\right)...\mathbf{end\; module}\right)$

$\mathrm{Hypercenter}\left(\mathrm{DihedralGroup}\left(12\right)\right)$

$\mathrm{GroupTheory}:-\mathrm{PermutationGroup}\left(\left\{\mathbf{module}\left(\right)...\mathbf{end\; module}\&comma;\mathbf{module}\left(\right)...\mathbf{end\; module}\right\}\&comma;\mathrm{degree}\&equals;12\&comma;\mathrm{supergroup}\&equals;\mathrm{GroupTheory}:-\mathrm{PermutationGroup}\left(\left\{\mathbf{module}\left(\right)...\mathbf{end\; module}\&comma;\mathbf{module}\left(\right)...\mathbf{end\; module}\right\}\&comma;\mathrm{degree}\&equals;12\right)\right)$

$\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left(42^k\right)\right)\mathbf{assuming}k::\&apos;\mathrm{posint}\&apos;$

$\mathrm{true}$

$\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left(6^k\right)\right)\mathbf{assuming}k::\&apos;\mathrm{posint}\&apos;$

$\mathrm{false}$

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

For more information on Maple 17 changes, see Updates in Maple 17.