Orbit - Maple Help

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

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{RubiksCubeGroup}\left(\right)$
 ${G}{≔}{\mathrm{< a permutation group on 48 letters with 6 generators >}}$ (1)
 > $\mathrm{O1}≔\mathrm{Orbit}\left(1,G\right)$
 ${\mathrm{O1}}{≔}{{1}}^{{\mathrm{< a permutation group on 48 letters with 6 generators >}}}$ (2)
 > $\mathrm{numelems}\left(\mathrm{O1}\right)$
 ${24}$ (3)
 > $2\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}∈\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{O1}$
 ${\mathrm{false}}$ (4)
 > $\mathrm{O2}≔\mathrm{Orbit}\left(2,G\right)$
 ${\mathrm{O2}}{≔}{{2}}^{{\mathrm{< a permutation group on 48 letters with 6 generators >}}}$ (5)
 > $\mathrm{numelems}\left(\mathrm{O2}\right)$
 ${24}$ (6)
 > $\mathrm{orbs}≔\mathrm{Orbits}\left(G\right)$
 ${\mathrm{orbs}}{≔}\left[{{1}}^{{\mathrm{< a permutation group on 48 letters with 6 generators >}}}{,}{{2}}^{{\mathrm{< a permutation group on 48 letters with 6 generators >}}}\right]$ (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.