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}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{15}\right)\left({4}{,}{11}\right)\left({5}{,}{10}\right)\left({6}{,}{12}\right)\left({7}{,}{14}\right)\left({8}{,}{13}\right)\left({9}{,}{16}\right)\left({17}{,}{28}\right)\left({18}{,}{27}\right)\left({19}{,}{38}\right)\left({20}{,}{37}\right)\left({21}{,}{34}\right)\left({22}{,}{33}\right)\left({23}{,}{32}\right)\left({24}{,}{31}\right)\left({25}{,}{36}\right)\left({26}{,}{35}\right)\left({29}{,}{40}\right)\left({30}{,}{39}\right)\left({41}{,}{48}\right)\left({42}{,}{47}\right)\left({43}{,}{46}\right)\left({44}{,}{45}\right){,}\left({1}{,}{3}\right)\left({2}{,}{9}\right)\left({4}{,}{17}\right)\left({5}{,}{16}\right)\left({6}{,}{18}\right)\left({7}{,}{19}\right)\left({8}{,}{20}\right)\left({10}{,}{27}\right)\left({11}{,}{15}\right)\left({12}{,}{28}\right)\left({13}{,}{29}\right)\left({14}{,}{30}\right)\left({21}{,}{41}\right)\left({22}{,}{42}\right)\left({23}{,}{39}\right)\left({24}{,}{40}\right)\left({25}{,}{43}\right)\left({26}{,}{44}\right)\left({31}{,}{45}\right)\left({32}{,}{46}\right)\left({33}{,}{37}\right)\left({34}{,}{38}\right)\left({35}{,}{47}\right)\left({36}{,}{48}\right){,}\left({1}{,}{4}{,}{6}{,}{5}\right)\left({2}{,}{10}{,}{12}{,}{11}\right)\left({3}{,}{16}{,}{18}{,}{17}\right)\left({7}{,}{21}{,}{25}{,}{23}\right)\left({8}{,}{22}{,}{26}{,}{24}\right)\left({9}{,}{15}{,}{28}{,}{27}\right)\left({13}{,}{31}{,}{35}{,}{33}\right)\left({14}{,}{32}{,}{36}{,}{34}\right)\left({19}{,}{39}{,}{43}{,}{41}\right)\left({20}{,}{40}{,}{44}{,}{42}\right)\left({29}{,}{37}{,}{47}{,}{45}\right)\left({30}{,}{38}{,}{48}{,}{46}\right){,}\left({1}{,}{6}\right)\left({2}{,}{12}\right)\left({3}{,}{18}\right)\left({4}{,}{5}\right)\left({7}{,}{25}\right)\left({8}{,}{26}\right)\left({9}{,}{28}\right)\left({10}{,}{11}\right)\left({13}{,}{35}\right)\left({14}{,}{36}\right)\left({15}{,}{27}\right)\left({16}{,}{17}\right)\left({19}{,}{43}\right)\left({20}{,}{44}\right)\left({21}{,}{23}\right)\left({22}{,}{24}\right)\left({29}{,}{47}\right)\left({30}{,}{48}\right)\left({31}{,}{33}\right)\left({32}{,}{34}\right)\left({37}{,}{45}\right)\left({38}{,}{46}\right)\left({39}{,}{41}\right)\left({40}{,}{42}\right){,}\left({1}{,}{7}{,}{8}\right)\left({2}{,}{13}{,}{14}\right)\left({3}{,}{19}{,}{20}\right)\left({4}{,}{21}{,}{22}\right)\left({5}{,}{23}{,}{24}\right)\left({6}{,}{25}{,}{26}\right)\left({9}{,}{29}{,}{30}\right)\left({10}{,}{31}{,}{32}\right)\left({11}{,}{33}{,}{34}\right)\left({12}{,}{35}{,}{36}\right)\left({15}{,}{37}{,}{38}\right)\left({16}{,}{39}{,}{40}\right)\left({17}{,}{41}{,}{42}\right)\left({18}{,}{43}{,}{44}\right)\left({27}{,}{45}{,}{46}\right)\left({28}{,}{47}{,}{48}\right)⟩$ (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)