AgemoPGroup - Maple Help

GroupTheory

 AgemoPGroup
 construct an Agemo of a p-group
 OmegaPGroup
 construct an Omega of a p-group

 Calling Sequence AgemoPGroup( G ) AgemoPGroup( n, G ) OmegaPGroup( G ) OmegaPGroup( n, G )

Parameters

 G - : PermutationGroup; a permutation $p$-group, for a prime number $p$ n - : nonnegint; (optional) a non-negative integer, default $n=1$

Description

 • If $n$ is a non-negative integer, and $G$ is a finite $p$-group, then the subgroup ${\mathrm{℧}}_{n}\left(G\right)$ is defined to be the subgroup of $G$ generated by elements of $G$ of the form ${g}^{{p}^{n}}$, as $g$ ranges over all elements of $G$.
 • The AgemoPGroup( n, G ) command computes the subgroup ${\mathrm{℧}}_{n}\left(G\right)$ of G, where G is a permutation $p$-group, for some prime $p$.
 • The first argument n is optional and is equal to $1$ by default. That is, the command AgemoPGroup( G ) is equivalent to AgemoPGroup( 1, G ).
 • For a $p$-group $G$, and a non-negative integer $n$, the subgroup ${\mathrm{Ω}}_{n}\left(G\right)$ is defined to be the subgroup generated by the elements $g$ such that ${g}^{{p}^{n}}$ = 1, for $g\in G$. That is, the subgroup generated by those members of $G$ whose order divides ${p}^{n}$.
 • The OmegaPGroup( n, G ) command computes ${\mathrm{Ω}}_{n}\left(G\right)$ for a permutation group G of prime power order.
 • When called with two arguments, $n$ and $G$, the indicated subgroup ${\mathrm{Ω}}_{n}\left(G\right)$ is returned. When called with just one argument $G$, the subgroup ${\mathrm{Ω}}_{1}\left(G\right)$ is returned.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{DihedralGroup}\left(8\right)$
 ${G}{≔}{{\mathbf{D}}}_{{8}}$ (1)
 > $A≔\mathrm{AgemoPGroup}\left(G\right)$
 ${A}{≔}{{&Agemo;}}_{{1}}{}\left({{\mathbf{D}}}_{{8}}\right)$ (2)
 > $\mathrm{IsCyclic}\left(A\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{GroupOrder}\left(A\right)$
 ${4}$ (4)
 > $A≔\mathrm{AgemoPGroup}\left(2,G\right)$
 ${A}{≔}{{&Agemo;}}_{{2}}{}\left({{\mathbf{D}}}_{{8}}\right)$ (5)
 > $\mathrm{GroupOrder}\left(A\right)$
 ${2}$ (6)
 > $\mathrm{AgemoPGroup}\left(0,G\right)$
 ${{\mathbf{D}}}_{{8}}$ (7)
 > $G≔\mathrm{CyclicGroup}\left(16807\right)$
 ${G}{≔}{{C}}_{{16807}}$ (8)
 > $\mathrm{seq}\left(\mathrm{GroupOrder}\left(\mathrm{AgemoPGroup}\left(n,G\right)\right),n=0..5\right)$
 ${16807}{,}{2401}{,}{343}{,}{49}{,}{7}{,}{1}$ (9)
 > $G≔\mathrm{QuaternionGroup}\left(5\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}{,}{9}{,}{10}{,}{11}{,}{12}{,}{13}{,}{14}{,}{15}{,}{16}\right)\left({17}{,}{18}{,}{19}{,}{20}{,}{21}{,}{22}{,}{23}{,}{24}{,}{25}{,}{26}{,}{27}{,}{28}{,}{29}{,}{30}{,}{31}{,}{32}\right){,}\left({1}{,}{31}{,}{9}{,}{23}\right)\left({2}{,}{30}{,}{10}{,}{22}\right)\left({3}{,}{29}{,}{11}{,}{21}\right)\left({4}{,}{28}{,}{12}{,}{20}\right)\left({5}{,}{27}{,}{13}{,}{19}\right)\left({6}{,}{26}{,}{14}{,}{18}\right)\left({7}{,}{25}{,}{15}{,}{17}\right)\left({8}{,}{24}{,}{16}{,}{32}\right)⟩$ (10)
 > $W≔\mathrm{OmegaPGroup}\left(G\right)$
 ${W}{≔}{{\Omega }}_{{1}}{}\left(⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}{,}{9}{,}{10}{,}{11}{,}{12}{,}{13}{,}{14}{,}{15}{,}{16}\right)\left({17}{,}{18}{,}{19}{,}{20}{,}{21}{,}{22}{,}{23}{,}{24}{,}{25}{,}{26}{,}{27}{,}{28}{,}{29}{,}{30}{,}{31}{,}{32}\right){,}\left({1}{,}{31}{,}{9}{,}{23}\right)\left({2}{,}{30}{,}{10}{,}{22}\right)\left({3}{,}{29}{,}{11}{,}{21}\right)\left({4}{,}{28}{,}{12}{,}{20}\right)\left({5}{,}{27}{,}{13}{,}{19}\right)\left({6}{,}{26}{,}{14}{,}{18}\right)\left({7}{,}{25}{,}{15}{,}{17}\right)\left({8}{,}{24}{,}{16}{,}{32}\right)⟩\right)$ (11)
 > $\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(2,G\right)\right)$
 ${32}$ (12)
 > $G≔\mathrm{CyclicGroup}\left(625\right)$
 ${G}{≔}{{C}}_{{625}}$ (13)
 > $\mathrm{seq}\left(\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(n,G\right)\right),n=0..4\right)$
 ${1}{,}{5}{,}{25}{,}{125}{,}{625}$ (14)

While it is immediate from the definition that ${\mathrm{Ω}}_{n}\left(G\right)\le {\mathrm{Ω}}_{n+1}\left(G\right)$, for all $n$ and any finite $p$-group $G$, equality may occur.

 > $G≔\mathrm{SmallGroup}\left(32,38\right):$
 > $\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(1,G\right)\right)$
 ${16}$ (15)
 > $\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(2,G\right)\right)$
 ${16}$ (16)

However, we must eventually reach the entire group $G$.

 > $\mathrm{GroupOrder}\left(G\right)=\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(3,G\right)\right)$
 ${32}{=}{32}$ (17)
 > $G≔\mathrm{WreathProduct}\left(\mathrm{SmallGroup}\left(27,4\right),\mathrm{CyclicGroup}\left(3\right)\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{8}{,}{6}{,}{11}{,}{15}{,}{7}{,}{12}{,}{3}\right)\left({4}{,}{16}{,}{13}{,}{18}{,}{25}{,}{24}{,}{19}{,}{9}{,}{17}\right)\left({5}{,}{20}{,}{27}{,}{22}{,}{10}{,}{14}{,}{23}{,}{26}{,}{21}\right){,}\left({1}{,}{4}{,}{5}\right)\left({2}{,}{9}{,}{10}\right)\left({3}{,}{13}{,}{14}\right)\left({6}{,}{18}{,}{22}\right)\left({7}{,}{19}{,}{23}\right)\left({8}{,}{24}{,}{21}\right)\left({11}{,}{16}{,}{26}\right)\left({12}{,}{25}{,}{20}\right)\left({15}{,}{17}{,}{27}\right){,}\left({1}{,}{6}{,}{7}\right)\left({2}{,}{11}{,}{12}\right)\left({3}{,}{8}{,}{15}\right)\left({4}{,}{18}{,}{19}\right)\left({5}{,}{22}{,}{23}\right)\left({9}{,}{16}{,}{25}\right)\left({10}{,}{26}{,}{20}\right)\left({13}{,}{24}{,}{17}\right)\left({14}{,}{21}{,}{27}\right){,}\left({1}{,}{28}{,}{55}\right)\left({2}{,}{29}{,}{56}\right)\left({3}{,}{30}{,}{57}\right)\left({4}{,}{31}{,}{58}\right)\left({5}{,}{32}{,}{59}\right)\left({6}{,}{33}{,}{60}\right)\left({7}{,}{34}{,}{61}\right)\left({8}{,}{35}{,}{62}\right)\left({9}{,}{36}{,}{63}\right)\left({10}{,}{37}{,}{64}\right)\left({11}{,}{38}{,}{65}\right)\left({12}{,}{39}{,}{66}\right)\left({13}{,}{40}{,}{67}\right)\left({14}{,}{41}{,}{68}\right)\left({15}{,}{42}{,}{69}\right)\left({16}{,}{43}{,}{70}\right)\left({17}{,}{44}{,}{71}\right)\left({18}{,}{45}{,}{72}\right)\left({19}{,}{46}{,}{73}\right)\left({20}{,}{47}{,}{74}\right)\left({21}{,}{48}{,}{75}\right)\left({22}{,}{49}{,}{76}\right)\left({23}{,}{50}{,}{77}\right)\left({24}{,}{51}{,}{78}\right)\left({25}{,}{52}{,}{79}\right)\left({26}{,}{53}{,}{80}\right)\left({27}{,}{54}{,}{81}\right)⟩$ (18)
 > $W≔\mathrm{OmegaPGroup}\left(G\right)$
 ${W}{≔}{{\Omega }}_{{1}}{}\left(⟨\left({1}{,}{2}{,}{8}{,}{6}{,}{11}{,}{15}{,}{7}{,}{12}{,}{3}\right)\left({4}{,}{16}{,}{13}{,}{18}{,}{25}{,}{24}{,}{19}{,}{9}{,}{17}\right)\left({5}{,}{20}{,}{27}{,}{22}{,}{10}{,}{14}{,}{23}{,}{26}{,}{21}\right){,}\left({1}{,}{4}{,}{5}\right)\left({2}{,}{9}{,}{10}\right)\left({3}{,}{13}{,}{14}\right)\left({6}{,}{18}{,}{22}\right)\left({7}{,}{19}{,}{23}\right)\left({8}{,}{24}{,}{21}\right)\left({11}{,}{16}{,}{26}\right)\left({12}{,}{25}{,}{20}\right)\left({15}{,}{17}{,}{27}\right){,}\left({1}{,}{6}{,}{7}\right)\left({2}{,}{11}{,}{12}\right)\left({3}{,}{8}{,}{15}\right)\left({4}{,}{18}{,}{19}\right)\left({5}{,}{22}{,}{23}\right)\left({9}{,}{16}{,}{25}\right)\left({10}{,}{26}{,}{20}\right)\left({13}{,}{24}{,}{17}\right)\left({14}{,}{21}{,}{27}\right){,}\left({1}{,}{28}{,}{55}\right)\left({2}{,}{29}{,}{56}\right)\left({3}{,}{30}{,}{57}\right)\left({4}{,}{31}{,}{58}\right)\left({5}{,}{32}{,}{59}\right)\left({6}{,}{33}{,}{60}\right)\left({7}{,}{34}{,}{61}\right)\left({8}{,}{35}{,}{62}\right)\left({9}{,}{36}{,}{63}\right)\left({10}{,}{37}{,}{64}\right)\left({11}{,}{38}{,}{65}\right)\left({12}{,}{39}{,}{66}\right)\left({13}{,}{40}{,}{67}\right)\left({14}{,}{41}{,}{68}\right)\left({15}{,}{42}{,}{69}\right)\left({16}{,}{43}{,}{70}\right)\left({17}{,}{44}{,}{71}\right)\left({18}{,}{45}{,}{72}\right)\left({19}{,}{46}{,}{73}\right)\left({20}{,}{47}{,}{74}\right)\left({21}{,}{48}{,}{75}\right)\left({22}{,}{49}{,}{76}\right)\left({23}{,}{50}{,}{77}\right)\left({24}{,}{51}{,}{78}\right)\left({25}{,}{52}{,}{79}\right)\left({26}{,}{53}{,}{80}\right)\left({27}{,}{54}{,}{81}\right)⟩\right)$ (19)
 > $\mathrm{GroupOrder}\left(W\right)$
 ${19683}$ (20)
 > $\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(2,G\right)\right)$
 ${59049}$ (21)
 > $G≔\mathrm{DirectProduct}\left(\mathrm{}\left(\mathrm{QuaternionGroup}\left(\right),4\right),\mathrm{CyclicGroup}\left(4\right),\mathrm{}\left(\mathrm{DihedralGroup}\left(16\right),3\right)\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}{,}{4}\right)\left({5}{,}{6}{,}{8}{,}{7}\right){,}\left({1}{,}{5}{,}{3}{,}{8}\right)\left({2}{,}{7}{,}{4}{,}{6}\right){,}\left({9}{,}{10}{,}{11}{,}{12}\right)\left({13}{,}{14}{,}{16}{,}{15}\right){,}\left({9}{,}{13}{,}{11}{,}{16}\right)\left({10}{,}{15}{,}{12}{,}{14}\right){,}\left({17}{,}{18}{,}{19}{,}{20}\right)\left({21}{,}{22}{,}{24}{,}{23}\right){,}\left({17}{,}{21}{,}{19}{,}{24}\right)\left({18}{,}{23}{,}{20}{,}{22}\right){,}\left({25}{,}{26}{,}{27}{,}{28}\right)\left({29}{,}{30}{,}{32}{,}{31}\right){,}\left({25}{,}{29}{,}{27}{,}{32}\right)\left({26}{,}{31}{,}{28}{,}{30}\right){,}\left({33}{,}{34}{,}{35}{,}{36}\right){,}\left({37}{,}{38}{,}{39}{,}{40}{,}{41}{,}{42}{,}{43}{,}{44}{,}{45}{,}{46}{,}{47}{,}{48}{,}{49}{,}{50}{,}{51}{,}{52}\right){,}\left({37}{,}{51}\right)\left({38}{,}{50}\right)\left({39}{,}{49}\right)\left({40}{,}{48}\right)\left({41}{,}{47}\right)\left({42}{,}{46}\right)\left({43}{,}{45}\right){,}\left({53}{,}{54}{,}{55}{,}{56}{,}{57}{,}{58}{,}{59}{,}{60}{,}{61}{,}{62}{,}{63}{,}{64}{,}{65}{,}{66}{,}{67}{,}{68}\right){,}\left({53}{,}{67}\right)\left({54}{,}{66}\right)\left({55}{,}{65}\right)\left({56}{,}{64}\right)\left({57}{,}{63}\right)\left({58}{,}{62}\right)\left({59}{,}{61}\right){,}\left({69}{,}{70}{,}{71}{,}{72}{,}{73}{,}{74}{,}{75}{,}{76}{,}{77}{,}{78}{,}{79}{,}{80}{,}{81}{,}{82}{,}{83}{,}{84}\right){,}\left({69}{,}{83}\right)\left({70}{,}{82}\right)\left({71}{,}{81}\right)\left({72}{,}{80}\right)\left({73}{,}{79}\right)\left({74}{,}{78}\right)\left({75}{,}{77}\right)⟩$ (22)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${536870912}$ (23)
 > $\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(1,G\right)\right)$
 ${1048576}$ (24)
 > $\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(2,G\right)\right)$
 ${536870912}$ (25)