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GroupTheory

 LowerPCentralSeries
 construct the lower p-central series of a group

 Calling Sequence LowerPCentralSeries( p, G )

Parameters

 p - a prime number G - a permutation group

Description

 • 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 commutator subgroups, defined as follows. Let ${G}_{0}=G$ and, for $0, define ${G}_{k}={G}^{{p}_{G,{G}_{k-1}}}$. The sequence

$G={G}_{0}▹{G}_{1}▹\dots ▹{G}_{c}$

 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].

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{PermutationGroup}\left(\left\{\left[\left[1,2\right]\right],\left[\left[1,2,3\right],\left[4,5\right]\right]\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩$ (1)
 > $\mathrm{LowerPCentralSeries}\left(2,G\right)$
 $⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩{▹}⟨\left({1}{,}{3}{,}{2}\right)⟩$ (2)
 > $\mathrm{LowerPCentralSeries}\left(3,G\right)$
 $⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩$ (3)
 > $\mathrm{LowerPCentralSeries}\left(2,\mathrm{QuaternionGroup}\left(\right)\right)$
 ${Q}{▹}⟨\left({1}{,}{3}\right)\left({2}{,}{4}\right)\left({5}{,}{8}\right)\left({6}{,}{7}\right)⟩{▹}⟨⟩$ (4)
 > $\mathrm{LowerPCentralSeries}\left(2,\mathrm{DihedralGroup}\left(4\right)\right)$
 ${{\mathrm{D}}}_{{4}}{▹}⟨\left({1}{,}{3}\right)\left({2}{,}{4}\right)⟩{▹}⟨⟩$ (5)
 > $\mathrm{LowerPCentralSeries}\left(2,\mathrm{DihedralGroup}\left(5\right)\right)$
 ${{\mathrm{D}}}_{{5}}{▹}⟨\left({1}{,}{3}{,}{5}{,}{2}{,}{4}\right){,}\left({1}{,}{4}{,}{2}{,}{5}{,}{3}\right)⟩$ (6)

Compatibility

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