GroupTheory/AgemoSeries - Maple Help

GroupTheory

 AgemoSeries
 construct the agemo series of a _p_-group
 OmegaSeries
 construct the omega series of a _p_-group

 Calling Sequence AgemoSeries( G ) OmegaSeries( G )

Parameters

 G - a permutation group

Description

 • The agemo series of a $p$-group $G$, where $p$ is a prime number, is the descending normal series

$G={\mho }_{0}\left(G\right)▹{\mho }_{1}\left(G\right)▹\dots ▹{\mho }_{r}\left(G\right)=1$

of $G$ whose terms are the successive agemo subgroups ${\mathrm{℧}}_{n}\left(G\right)$ of $G$, where ${\mathrm{℧}}_{0}\left(G\right)=G$. See GroupTheory[AgemoPGroup]

 • The AgemoSeries( G ) command constructs the agemo series of a group G, which must be a finite $p$-group, for some prime $p$.
 • The omega series of a finite $p$-group $G$ is the ascending normal series

$1={\mathrm{\Omega }}_{0}\left(G\right)▹{\mathrm{\Omega }}_{1}\left(G\right)▹\dots ▹{\mathrm{\Omega }}_{r}\left(G\right)=G$

of $G$, whose terms are the successive omega subgroups ${\mathrm{Ω}}_{n}\left(G\right)$ of $G$. See GroupTheory[OmegaPGroup].

 • The OmegaSeries( G ) command constructs the omega series of a finite $p$-group G.
 • The group G must be an instance of a permutation group.
 • Both the agemo and omega series of G are represented by a NormalSeries object which admits certain operations common to all normal series.  See GroupTheory[Series].

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{DihedralGroup}\left(8\right)$
 ${G}{≔}{{\mathbf{D}}}_{{8}}$ (1)
 > $\mathrm{as}≔\mathrm{AgemoSeries}\left(G\right)$
 ${\mathrm{as}}{≔}{{\mathbf{D}}}_{{8}}{◃}{{&Agemo;}}_{{1}}{}\left({{\mathbf{D}}}_{{8}}\right){◃}{{&Agemo;}}_{{2}}{}\left({{\mathbf{D}}}_{{8}}\right){◃}{{&Agemo;}}_{{3}}{}\left({{\mathbf{D}}}_{{8}}\right)$ (2)
 > $\mathrm{numelems}\left(\mathrm{as}\right)$
 ${4}$ (3)
 > $\mathrm{os}≔\mathrm{OmegaSeries}\left(G\right)$
 ${\mathrm{os}}{≔}⟨⟩{◃}{{\mathbf{D}}}_{{8}}$ (4)
 > $\mathrm{numelems}\left(\mathrm{os}\right)$
 ${2}$ (5)