Orbit - Maple Help
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GroupTheory

 Orbit
 compute the orbit of a point under the action of a permutation group
 Orbits
 compute all the orbits of a permutation group

 Calling Sequence Orbit( alpha, G ) Orbits( G )

Parameters

 G - a permutation group alpha - posint; a point whose orbit is to be computed

Description

 • The Orbit( alpha, G ) command returns the orbit of the point alpha under the action of the permutation group G.
 • The returned value is an object that supports the following methods.

 Representative( orb ) returns a representative of the orbit orb numelems( orb ) returns the cardinality of the orbit orb member( x, orb ) or x in orb returns true if x belongs to the orbit orb Elements( orb ) returns the elements of the orbit orb, as a set

 • The Orbits( G ) command returns the set of all orbits of the permutation group G.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{RubiksCubeGroup}\left(\right)$
 ${G}{≔}⟨\left({6}{,}{25}{,}{43}{,}{16}\right)\left({7}{,}{28}{,}{42}{,}{13}\right)\left({8}{,}{30}{,}{41}{,}{11}\right)\left({17}{,}{19}{,}{24}{,}{22}\right)\left({18}{,}{21}{,}{23}{,}{20}\right){,}\left({1}{,}{14}{,}{48}{,}{27}\right)\left({2}{,}{12}{,}{47}{,}{29}\right)\left({3}{,}{9}{,}{46}{,}{32}\right)\left({33}{,}{35}{,}{40}{,}{38}\right)\left({34}{,}{37}{,}{39}{,}{36}\right){,}\left({1}{,}{17}{,}{41}{,}{40}\right)\left({4}{,}{20}{,}{44}{,}{37}\right)\left({6}{,}{22}{,}{46}{,}{35}\right)\left({9}{,}{11}{,}{16}{,}{14}\right)\left({10}{,}{13}{,}{15}{,}{12}\right){,}\left({3}{,}{38}{,}{43}{,}{19}\right)\left({5}{,}{36}{,}{45}{,}{21}\right)\left({8}{,}{33}{,}{48}{,}{24}\right)\left({25}{,}{27}{,}{32}{,}{30}\right)\left({26}{,}{29}{,}{31}{,}{28}\right){,}\left({1}{,}{3}{,}{8}{,}{6}\right)\left({2}{,}{5}{,}{7}{,}{4}\right)\left({9}{,}{33}{,}{25}{,}{17}\right)\left({10}{,}{34}{,}{26}{,}{18}\right)\left({11}{,}{35}{,}{27}{,}{19}\right){,}\left({14}{,}{22}{,}{30}{,}{38}\right)\left({15}{,}{23}{,}{31}{,}{39}\right)\left({16}{,}{24}{,}{32}{,}{40}\right)\left({41}{,}{43}{,}{48}{,}{46}\right)\left({42}{,}{45}{,}{47}{,}{44}\right)⟩$ (1)
 > $\mathrm{O1}≔\mathrm{Orbit}\left(1,G\right)$
 ${\mathrm{O1}}{≔}{{1}}^{⟨\left({6}{,}{25}{,}{43}{,}{16}\right)\left({7}{,}{28}{,}{42}{,}{13}\right)\left({8}{,}{30}{,}{41}{,}{11}\right)\left({17}{,}{19}{,}{24}{,}{22}\right)\left({18}{,}{21}{,}{23}{,}{20}\right){,}\left({1}{,}{14}{,}{48}{,}{27}\right)\left({2}{,}{12}{,}{47}{,}{29}\right)\left({3}{,}{9}{,}{46}{,}{32}\right)\left({33}{,}{35}{,}{40}{,}{38}\right)\left({34}{,}{37}{,}{39}{,}{36}\right){,}\left({1}{,}{17}{,}{41}{,}{40}\right)\left({4}{,}{20}{,}{44}{,}{37}\right)\left({6}{,}{22}{,}{46}{,}{35}\right)\left({9}{,}{11}{,}{16}{,}{14}\right)\left({10}{,}{13}{,}{15}{,}{12}\right){,}\left({3}{,}{38}{,}{43}{,}{19}\right)\left({5}{,}{36}{,}{45}{,}{21}\right)\left({8}{,}{33}{,}{48}{,}{24}\right)\left({25}{,}{27}{,}{32}{,}{30}\right)\left({26}{,}{29}{,}{31}{,}{28}\right){,}\left({1}{,}{3}{,}{8}{,}{6}\right)\left({2}{,}{5}{,}{7}{,}{4}\right)\left({9}{,}{33}{,}{25}{,}{17}\right)\left({10}{,}{34}{,}{26}{,}{18}\right)\left({11}{,}{35}{,}{27}{,}{19}\right){,}\left({14}{,}{22}{,}{30}{,}{38}\right)\left({15}{,}{23}{,}{31}{,}{39}\right)\left({16}{,}{24}{,}{32}{,}{40}\right)\left({41}{,}{43}{,}{48}{,}{46}\right)\left({42}{,}{45}{,}{47}{,}{44}\right)⟩}$ (2)
 > $\mathrm{numelems}\left(\mathrm{O1}\right)$
 ${24}$ (3)
 > $2\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{O1}$
 ${\mathrm{false}}$ (4)
 > $\mathrm{O2}≔\mathrm{Orbit}\left(2,G\right)$
 ${\mathrm{O2}}{≔}{{2}}^{⟨\left({6}{,}{25}{,}{43}{,}{16}\right)\left({7}{,}{28}{,}{42}{,}{13}\right)\left({8}{,}{30}{,}{41}{,}{11}\right)\left({17}{,}{19}{,}{24}{,}{22}\right)\left({18}{,}{21}{,}{23}{,}{20}\right){,}\left({1}{,}{14}{,}{48}{,}{27}\right)\left({2}{,}{12}{,}{47}{,}{29}\right)\left({3}{,}{9}{,}{46}{,}{32}\right)\left({33}{,}{35}{,}{40}{,}{38}\right)\left({34}{,}{37}{,}{39}{,}{36}\right){,}\left({1}{,}{17}{,}{41}{,}{40}\right)\left({4}{,}{20}{,}{44}{,}{37}\right)\left({6}{,}{22}{,}{46}{,}{35}\right)\left({9}{,}{11}{,}{16}{,}{14}\right)\left({10}{,}{13}{,}{15}{,}{12}\right){,}\left({3}{,}{38}{,}{43}{,}{19}\right)\left({5}{,}{36}{,}{45}{,}{21}\right)\left({8}{,}{33}{,}{48}{,}{24}\right)\left({25}{,}{27}{,}{32}{,}{30}\right)\left({26}{,}{29}{,}{31}{,}{28}\right){,}\left({1}{,}{3}{,}{8}{,}{6}\right)\left({2}{,}{5}{,}{7}{,}{4}\right)\left({9}{,}{33}{,}{25}{,}{17}\right)\left({10}{,}{34}{,}{26}{,}{18}\right)\left({11}{,}{35}{,}{27}{,}{19}\right){,}\left({14}{,}{22}{,}{30}{,}{38}\right)\left({15}{,}{23}{,}{31}{,}{39}\right)\left({16}{,}{24}{,}{32}{,}{40}\right)\left({41}{,}{43}{,}{48}{,}{46}\right)\left({42}{,}{45}{,}{47}{,}{44}\right)⟩}$ (5)
 > $\mathrm{numelems}\left(\mathrm{O2}\right)$
 ${24}$ (6)
 > $\mathrm{orbs}≔\mathrm{Orbits}\left(G\right)$
 ${\mathrm{orbs}}{≔}\left[{{1}}^{⟨\left({6}{,}{25}{,}{43}{,}{16}\right)\left({7}{,}{28}{,}{42}{,}{13}\right)\left({8}{,}{30}{,}{41}{,}{11}\right)\left({17}{,}{19}{,}{24}{,}{22}\right)\left({18}{,}{21}{,}{23}{,}{20}\right){,}\left({1}{,}{14}{,}{48}{,}{27}\right)\left({2}{,}{12}{,}{47}{,}{29}\right)\left({3}{,}{9}{,}{46}{,}{32}\right)\left({33}{,}{35}{,}{40}{,}{38}\right)\left({34}{,}{37}{,}{39}{,}{36}\right){,}\left({1}{,}{17}{,}{41}{,}{40}\right)\left({4}{,}{20}{,}{44}{,}{37}\right)\left({6}{,}{22}{,}{46}{,}{35}\right)\left({9}{,}{11}{,}{16}{,}{14}\right)\left({10}{,}{13}{,}{15}{,}{12}\right){,}\left({3}{,}{38}{,}{43}{,}{19}\right)\left({5}{,}{36}{,}{45}{,}{21}\right)\left({8}{,}{33}{,}{48}{,}{24}\right)\left({25}{,}{27}{,}{32}{,}{30}\right)\left({26}{,}{29}{,}{31}{,}{28}\right){,}\left({1}{,}{3}{,}{8}{,}{6}\right)\left({2}{,}{5}{,}{7}{,}{4}\right)\left({9}{,}{33}{,}{25}{,}{17}\right)\left({10}{,}{34}{,}{26}{,}{18}\right)\left({11}{,}{35}{,}{27}{,}{19}\right){,}\left({14}{,}{22}{,}{30}{,}{38}\right)\left({15}{,}{23}{,}{31}{,}{39}\right)\left({16}{,}{24}{,}{32}{,}{40}\right)\left({41}{,}{43}{,}{48}{,}{46}\right)\left({42}{,}{45}{,}{47}{,}{44}\right)⟩}{,}{{2}}^{⟨\left({6}{,}{25}{,}{43}{,}{16}\right)\left({7}{,}{28}{,}{42}{,}{13}\right)\left({8}{,}{30}{,}{41}{,}{11}\right)\left({17}{,}{19}{,}{24}{,}{22}\right)\left({18}{,}{21}{,}{23}{,}{20}\right){,}\left({1}{,}{14}{,}{48}{,}{27}\right)\left({2}{,}{12}{,}{47}{,}{29}\right)\left({3}{,}{9}{,}{46}{,}{32}\right)\left({33}{,}{35}{,}{40}{,}{38}\right)\left({34}{,}{37}{,}{39}{,}{36}\right){,}\left({1}{,}{17}{,}{41}{,}{40}\right)\left({4}{,}{20}{,}{44}{,}{37}\right)\left({6}{,}{22}{,}{46}{,}{35}\right)\left({9}{,}{11}{,}{16}{,}{14}\right)\left({10}{,}{13}{,}{15}{,}{12}\right){,}\left({3}{,}{38}{,}{43}{,}{19}\right)\left({5}{,}{36}{,}{45}{,}{21}\right)\left({8}{,}{33}{,}{48}{,}{24}\right)\left({25}{,}{27}{,}{32}{,}{30}\right)\left({26}{,}{29}{,}{31}{,}{28}\right){,}\left({1}{,}{3}{,}{8}{,}{6}\right)\left({2}{,}{5}{,}{7}{,}{4}\right)\left({9}{,}{33}{,}{25}{,}{17}\right)\left({10}{,}{34}{,}{26}{,}{18}\right)\left({11}{,}{35}{,}{27}{,}{19}\right){,}\left({14}{,}{22}{,}{30}{,}{38}\right)\left({15}{,}{23}{,}{31}{,}{39}\right)\left({16}{,}{24}{,}{32}{,}{40}\right)\left({41}{,}{43}{,}{48}{,}{46}\right)\left({42}{,}{45}{,}{47}{,}{44}\right)⟩}\right]$ (7)
 > $\mathrm{nops}\left(\mathrm{orbs}\right)$
 ${2}$ (8)
 > $\mathrm{Elements}\left(\mathrm{O2}\right)$
 $\left\{{2}{,}{4}{,}{5}{,}{7}{,}{10}{,}{12}{,}{13}{,}{15}{,}{18}{,}{20}{,}{21}{,}{23}{,}{26}{,}{28}{,}{29}{,}{31}{,}{34}{,}{36}{,}{37}{,}{39}{,}{42}{,}{44}{,}{45}{,}{47}\right\}$ (9)

Compatibility

 • The GroupTheory[Orbit] and GroupTheory[Orbits] commands were introduced in Maple 17.
 • For more information on Maple 17 changes, see Updates in Maple 17.

 See Also