The lower p-central series of a group $G$, for a prime number $p$, is the descending normal series of $G$ whose terms are the successive subgroups, defined as follows. Let $G_0=G$ and, for $0<k$, define $G_k=G^p·\left[G\&comma;G_{k-1}\right]$. The sequence

is called the lower p-central series of $G$. If the $p$-residual $G_c$ is the trivial group, then $G$ is a $p$-group. In this case, the number $c$ is called the p- class of $G$.

The LowerPCentralSeries( G ) command constructs the lower $p$-central series of a group G.

The group G must be an instance of a permutation group.

The lower $p$-central series of G is represented by a series data structure which admits certain operations common to all series.  See GroupTheory[Series].

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

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

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

$\mathrm{LowerPCentralSeries}\left(2\&comma;G\right)$

$⟨\left(1\&comma;2\right)\&comma;\left(1\&comma;2\&comma;3\right)\left(4\&comma;5\right)⟩▹⟨\left(1\&comma;3\&comma;2\right)⟩$

$\mathrm{LowerPCentralSeries}\left(3\&comma;G\right)$

$⟨\left(1\&comma;2\right)\&comma;\left(1\&comma;2\&comma;3\right)\left(4\&comma;5\right)⟩$

$\mathrm{LowerPCentralSeries}\left(2\&comma;\mathrm{QuaternionGroup}\left(\right)\right)$

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

$\mathrm{LowerPCentralSeries}\left(2\&comma;\mathrm{DihedralGroup}\left(4\right)\right)$

${\mathbf{D}}_4▹⟨\left(1\&comma;3\right)\left(2\&comma;4\right)⟩▹⟨⟩$

$\mathrm{LowerPCentralSeries}\left(2\&comma;\mathrm{DihedralGroup}\left(5\right)\right)$

${\mathbf{D}}_5▹⟨\left(1\&comma;4\&comma;2\&comma;5\&comma;3\right)⟩$

The GroupTheory[LowerPCentralSeries] command was introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.