GroupTheory/ReducedDegreePermGroup - Maple Help

GroupTheory

 ReducedDegreePermGroup
 try to find an isomorphic permutation group of smaller degree

 Calling Sequence ReducedDegreePermGroup( G )

Parameters

 G - PermutationGroup; a permutation group

Description

 • The ReducedDegreePermGroup( G ) command returns a permutation group isomorphic (as an abstract group) with possibly smaller degree, if one can be found.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SmallGroup}\left(48,15\right)$
 ${G}{≔}{\mathrm{< a permutation group on 48 letters with 5 generators >}}$ (1)
 > $\mathrm{Degree}\left(G\right)$
 ${48}$ (2)
 > $R≔\mathrm{ReducedDegreePermGroup}\left(G\right)$
 ${R}{≔}⟨\left({2}{,}{8}\right)\left({3}{,}{4}\right)\left({6}{,}{7}\right)\left({9}{,}{10}\right)\left({11}{,}{20}\right)\left({12}{,}{19}\right)\left({13}{,}{16}\right)\left({14}{,}{15}\right)\left({17}{,}{18}\right)\left({21}{,}{24}\right)\left({22}{,}{23}\right){,}\left({1}{,}{2}\right)\left({3}{,}{9}\right)\left({4}{,}{8}\right)\left({5}{,}{10}\right)\left({6}{,}{11}\right)\left({7}{,}{12}\right)\left({13}{,}{21}\right)\left({14}{,}{22}\right)\left({15}{,}{19}\right)\left({16}{,}{20}\right)\left({17}{,}{23}\right)\left({18}{,}{24}\right){,}\left({1}{,}{6}{,}{7}\right)\left({2}{,}{11}{,}{12}\right)\left({3}{,}{13}{,}{14}\right)\left({4}{,}{15}{,}{16}\right)\left({5}{,}{17}{,}{18}\right)\left({8}{,}{19}{,}{20}\right)\left({9}{,}{21}{,}{22}\right)\left({10}{,}{23}{,}{24}\right)⟩$ (3)
 > $\mathrm{Degree}\left(R\right)$
 ${24}$ (4)

It is not always possible to produce an isomorphic permutation group with smaller degree.

 > $G≔\mathrm{CyclicGroup}\left(9\right)$
 ${G}{≔}{{C}}_{{9}}$ (5)
 > $\mathrm{Degree}\left(G\right)$
 ${9}$ (6)
 > $R≔\mathrm{ReducedDegreePermGroup}\left(G\right)$
 ${R}{≔}{{C}}_{{9}}$ (7)
 > $\mathrm{Degree}\left(R\right)$
 ${9}$ (8)

On the other hand, particularly for groups produced either from a finitely presented group (which are often regular), or via a linear or projective action on a vector space, the degree can be reduced substantially.

 > $G≔\mathrm{MathieuGroup}\left(11,'\mathrm{form}'="fpgroup"\right)$
 ${G}{≔}⟨{}{a}{,}{b}{}{\mid }{}{{a}}^{{2}}{,}{{b}}^{{4}}{,}{a}{}{{b}}^{{2}}{}{a}{}{{b}}^{{2}}{}{a}{}{{b}}^{{2}}{}{a}{}{{b}}^{{2}}{}{a}{}{{b}}^{{2}}{}{a}{}{{b}}^{{2}}{,}{a}{}{b}{}{a}{}{b}{}{a}{}{{b}}^{{-1}}{}{a}{}{b}{}{a}{}{{b}}^{{2}}{}{a}{}{{b}}^{{-1}}{}{a}{}{b}{}{a}{}{{b}}^{{-1}}{}{a}{}{{b}}^{{-1}}{,}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}{a}{}{b}{}⟩$ (9)
 > $P≔\mathrm{PermutationGroup}\left(G\right):$
 > $\mathrm{Degree}\left(P\right)$
 ${7920}$ (10)
 > $\mathrm{IsRegular}\left(P\right)$
 ${\mathrm{true}}$ (11)
 > $R≔\mathrm{ReducedDegreePermGroup}\left(P\right)$
 ${R}{≔}⟨\left({1}{,}{2}\right)\left({4}{,}{5}\right)\left({7}{,}{9}\right)\left({10}{,}{11}\right){,}\left({1}{,}{2}{,}{4}{,}{3}\right)\left({5}{,}{6}{,}{8}{,}{7}\right)\left({9}{,}{10}\right)\left({11}{,}{12}\right)⟩$ (12)
 > $\mathrm{Degree}\left(R\right)$
 ${12}$ (13)