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].

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

$G≔\mathrm{PermutationGroup}\left(\left[\mathrm{Perm}\left(\left[\left[1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\right]\right]\right)\&comma;\mathrm{Perm}\left(\left[\left[1\&comma;7\right]\&comma;\left[2\&comma;6\right]\&comma;\left[3\&comma;5\right]\right]\right)\right]\right)$

$G≔⟨\left(1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\right)\&comma;\left(1\&comma;7\right)\left(2\&comma;6\right)\left(3\&comma;5\right)⟩$

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

$⟨\left(1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\right)\&comma;\left(1\&comma;7\right)\left(2\&comma;6\right)\left(3\&comma;5\right)⟩▹\left[⟨\left(1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\right)\&comma;\left(1\&comma;7\right)\left(2\&comma;6\right)\left(3\&comma;5\right)⟩\&comma;⟨\left(1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\right)\&comma;\left(1\&comma;7\right)\left(2\&comma;6\right)\left(3\&comma;5\right)⟩\right]▹\left[⟨\left(1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\right)\&comma;\left(1\&comma;7\right)\left(2\&comma;6\right)\left(3\&comma;5\right)⟩\&comma;\left[⟨\left(1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\right)\&comma;\left(1\&comma;7\right)\left(2\&comma;6\right)\left(3\&comma;5\right)⟩\&comma;⟨\left(1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\right)\&comma;\left(1\&comma;7\right)\left(2\&comma;6\right)\left(3\&comma;5\right)⟩\right]\right]▹\left[⟨\left(1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\right)\&comma;\left(1\&comma;7\right)\left(2\&comma;6\right)\left(3\&comma;5\right)⟩\&comma;\left[⟨\left(1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\right)\&comma;\left(1\&comma;7\right)\left(2\&comma;6\right)\left(3\&comma;5\right)⟩\&comma;\left[⟨\left(1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\right)\&comma;\left(1\&comma;7\right)\left(2\&comma;6\right)\left(3\&comma;5\right)⟩\&comma;⟨\left(1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\right)\&comma;\left(1\&comma;7\right)\left(2\&comma;6\right)\left(3\&comma;5\right)⟩\right]\right]\right]$

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

Warning, over-writing property `["LowerCentralSeries"]' with a different value

$\mathrm{true}$

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

$3$

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

$\mathrm{lcs}≔{\mathbf{A}}_4▹\left[{\mathbf{A}}_4\&comma;{\mathbf{A}}_4\right]$

$\mathrm{type}\left(\mathrm{lcs}\&comma;'\mathrm{NormalSeries}'\right)$

$\mathrm{true}$

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

${\mathbf{A}}_4$

$\left[{\mathbf{A}}_4\&comma;{\mathbf{A}}_4\right]$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

$\left[{\mathbf{D}}_{12}\&comma;\left[{\mathbf{D}}_{12}\&comma;{\mathbf{D}}_{12}\right]\right]$

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

$⟨⟩▹⟨\left(1\&comma;9\right)\left(2\&comma;10\right)\left(3\&comma;11\right)\left(4\&comma;12\right)\left(5\&comma;13\right)\left(6\&comma;14\right)\left(7\&comma;15\right)\left(8\&comma;16\right)⟩▹\dots ▹⟨\left(1\&comma;9\right)\left(2\&comma;10\right)\left(3\&comma;11\right)\left(4\&comma;12\right)\left(5\&comma;13\right)\left(6\&comma;14\right)\left(7\&comma;15\right)\left(8\&comma;16\right)\&comma;\left(1\&comma;15\&comma;13\&comma;11\&comma;9\&comma;7\&comma;5\&comma;3\right)\left(2\&comma;16\&comma;14\&comma;12\&comma;10\&comma;8\&comma;6\&comma;4\right)\&comma;\left(1\&comma;7\&comma;13\&comma;3\&comma;9\&comma;15\&comma;5\&comma;11\right)\left(2\&comma;8\&comma;14\&comma;4\&comma;10\&comma;16\&comma;6\&comma;12\right)\&comma;\left(1\&comma;5\&comma;9\&comma;13\right)\left(2\&comma;6\&comma;10\&comma;14\right)\left(3\&comma;7\&comma;11\&comma;15\right)\left(4\&comma;8\&comma;12\&comma;16\right)⟩▹{\mathbf{D}}_{16}$

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

$⟨⟩▹⟨\left(1\&comma;7\right)\left(2\&comma;8\right)\left(3\&comma;9\right)\left(4\&comma;10\right)\left(5\&comma;11\right)\left(6\&comma;12\right)⟩▹⟨\left(1\&comma;10\&comma;7\&comma;4\right)\left(2\&comma;11\&comma;8\&comma;5\right)\left(3\&comma;12\&comma;9\&comma;6\right)\&comma;\left(1\&comma;7\right)\left(2\&comma;8\right)\left(3\&comma;9\right)\left(4\&comma;10\right)\left(5\&comma;11\right)\left(6\&comma;12\right)⟩$

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

$⟨\left(1\&comma;10\&comma;7\&comma;4\right)\left(2\&comma;11\&comma;8\&comma;5\right)\left(3\&comma;12\&comma;9\&comma;6\right)\&comma;\left(1\&comma;7\right)\left(2\&comma;8\right)\left(3\&comma;9\right)\left(4\&comma;10\right)\left(5\&comma;11\right)\left(6\&comma;12\right)⟩$

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

$\mathrm{true}$

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

$\mathrm{false}$

$\mathrm{seq}\left(\mathrm{NilpotencyClass}\left(\mathrm{QuaternionGroup}\left(n\right)\right)\&comma;n=3..10\right)$

$2,3,4,5,6,7,8,9$

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

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