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
Parameters
Description
Examples
Compatibility
Orbit( alpha, G )
Orbits( G )
G

a permutation group
alpha
posint; a point whose orbit is to be computed
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.
$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$
$G\u2254\mathrm{RubiksCubeGroup}\left(\right)$
${G}{\u2254}\u27e8\left({6}{\,}{25}{\,}{43}{\,}{16}\right)\left({7}{\,}{28}{\,}{42}{\,}{13}\right)\left({8}{\,}{30}{\,}{41}{\,}{11}\right)\left({17}{\,}{19}{\,}{24}{\,}{22}\right)\left({18}{\,}{21}{\,}{23}{\,}{20}\right){\,}\left({1}{\,}{14}{\,}{48}{\,}{27}\right)\left({2}{\,}{12}{\,}{47}{\,}{29}\right)\left({3}{\,}{9}{\,}{46}{\,}{32}\right)\left({33}{\,}{35}{\,}{40}{\,}{38}\right)\left({34}{\,}{37}{\,}{39}{\,}{36}\right){\,}\left({1}{\,}{17}{\,}{41}{\,}{40}\right)\left({4}{\,}{20}{\,}{44}{\,}{37}\right)\left({6}{\,}{22}{\,}{46}{\,}{35}\right)\left({9}{\,}{11}{\,}{16}{\,}{14}\right)\left({10}{\,}{13}{\,}{15}{\,}{12}\right){\,}\left({3}{\,}{38}{\,}{43}{\,}{19}\right)\left({5}{\,}{36}{\,}{45}{\,}{21}\right)\left({8}{\,}{33}{\,}{48}{\,}{24}\right)\left({25}{\,}{27}{\,}{32}{\,}{30}\right)\left({26}{\,}{29}{\,}{31}{\,}{28}\right){\,}\left({1}{\,}{3}{\,}{8}{\,}{6}\right)\left({2}{\,}{5}{\,}{7}{\,}{4}\right)\left({9}{\,}{33}{\,}{25}{\,}{17}\right)\left({10}{\,}{34}{\,}{26}{\,}{18}\right)\left({11}{\,}{35}{\,}{27}{\,}{19}\right){\,}\left({14}{\,}{22}{\,}{30}{\,}{38}\right)\left({15}{\,}{23}{\,}{31}{\,}{39}\right)\left({16}{\,}{24}{\,}{32}{\,}{40}\right)\left({41}{\,}{43}{\,}{48}{\,}{46}\right)\left({42}{\,}{45}{\,}{47}{\,}{44}\right)\u27e9$
$\mathrm{O1}\u2254\mathrm{Orbit}\left(1\,G\right)$
${\mathrm{O1}}{\u2254}{{1}}^{\u27e8\left({6}{\,}{25}{\,}{43}{\,}{16}\right)\left({7}{\,}{28}{\,}{42}{\,}{13}\right)\left({8}{\,}{30}{\,}{41}{\,}{11}\right)\left({17}{\,}{19}{\,}{24}{\,}{22}\right)\left({18}{\,}{21}{\,}{23}{\,}{20}\right){\,}\left({1}{\,}{14}{\,}{48}{\,}{27}\right)\left({2}{\,}{12}{\,}{47}{\,}{29}\right)\left({3}{\,}{9}{\,}{46}{\,}{32}\right)\left({33}{\,}{35}{\,}{40}{\,}{38}\right)\left({34}{\,}{37}{\,}{39}{\,}{36}\right){\,}\left({1}{\,}{17}{\,}{41}{\,}{40}\right)\left({4}{\,}{20}{\,}{44}{\,}{37}\right)\left({6}{\,}{22}{\,}{46}{\,}{35}\right)\left({9}{\,}{11}{\,}{16}{\,}{14}\right)\left({10}{\,}{13}{\,}{15}{\,}{12}\right){\,}\left({3}{\,}{38}{\,}{43}{\,}{19}\right)\left({5}{\,}{36}{\,}{45}{\,}{21}\right)\left({8}{\,}{33}{\,}{48}{\,}{24}\right)\left({25}{\,}{27}{\,}{32}{\,}{30}\right)\left({26}{\,}{29}{\,}{31}{\,}{28}\right){\,}\left({1}{\,}{3}{\,}{8}{\,}{6}\right)\left({2}{\,}{5}{\,}{7}{\,}{4}\right)\left({9}{\,}{33}{\,}{25}{\,}{17}\right)\left({10}{\,}{34}{\,}{26}{\,}{18}\right)\left({11}{\,}{35}{\,}{27}{\,}{19}\right){\,}\left({14}{\,}{22}{\,}{30}{\,}{38}\right)\left({15}{\,}{23}{\,}{31}{\,}{39}\right)\left({16}{\,}{24}{\,}{32}{\,}{40}\right)\left({41}{\,}{43}{\,}{48}{\,}{46}\right)\left({42}{\,}{45}{\,}{47}{\,}{44}\right)\u27e9}$
$\mathrm{numelems}\left(\mathrm{O1}\right)$
${24}$
$2\phantom{\rule[0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[0.0ex]{0.3em}{0.0ex}}\mathrm{O1}$
${\mathrm{false}}$
$\mathrm{O2}\u2254\mathrm{Orbit}\left(2\,G\right)$
${\mathrm{O2}}{\u2254}{{2}}^{\u27e8\left({6}{\,}{25}{\,}{43}{\,}{16}\right)\left({7}{\,}{28}{\,}{42}{\,}{13}\right)\left({8}{\,}{30}{\,}{41}{\,}{11}\right)\left({17}{\,}{19}{\,}{24}{\,}{22}\right)\left({18}{\,}{21}{\,}{23}{\,}{20}\right){\,}\left({1}{\,}{14}{\,}{48}{\,}{27}\right)\left({2}{\,}{12}{\,}{47}{\,}{29}\right)\left({3}{\,}{9}{\,}{46}{\,}{32}\right)\left({33}{\,}{35}{\,}{40}{\,}{38}\right)\left({34}{\,}{37}{\,}{39}{\,}{36}\right){\,}\left({1}{\,}{17}{\,}{41}{\,}{40}\right)\left({4}{\,}{20}{\,}{44}{\,}{37}\right)\left({6}{\,}{22}{\,}{46}{\,}{35}\right)\left({9}{\,}{11}{\,}{16}{\,}{14}\right)\left({10}{\,}{13}{\,}{15}{\,}{12}\right){\,}\left({3}{\,}{38}{\,}{43}{\,}{19}\right)\left({5}{\,}{36}{\,}{45}{\,}{21}\right)\left({8}{\,}{33}{\,}{48}{\,}{24}\right)\left({25}{\,}{27}{\,}{32}{\,}{30}\right)\left({26}{\,}{29}{\,}{31}{\,}{28}\right){\,}\left({1}{\,}{3}{\,}{8}{\,}{6}\right)\left({2}{\,}{5}{\,}{7}{\,}{4}\right)\left({9}{\,}{33}{\,}{25}{\,}{17}\right)\left({10}{\,}{34}{\,}{26}{\,}{18}\right)\left({11}{\,}{35}{\,}{27}{\,}{19}\right){\,}\left({14}{\,}{22}{\,}{30}{\,}{38}\right)\left({15}{\,}{23}{\,}{31}{\,}{39}\right)\left({16}{\,}{24}{\,}{32}{\,}{40}\right)\left({41}{\,}{43}{\,}{48}{\,}{46}\right)\left({42}{\,}{45}{\,}{47}{\,}{44}\right)\u27e9}$
$\mathrm{numelems}\left(\mathrm{O2}\right)$
$\mathrm{orbs}\u2254\mathrm{Orbits}\left(G\right)$
orbs≔16,25,43,167,28,42,138,30,41,1117,19,24,2218,21,23,20,1,14,48,272,12,47,293,9,46,3233,35,40,3834,37,39,36,1,17,41,404,20,44,376,22,46,359,11,16,1410,13,15,12,3,38,43,195,36,45,218,33,48,2425,27,32,3026,29,31,28,1,3,8,62,5,7,49,33,25,1710,34,26,1811,35,27,19,14,22,30,3815,23,31,3916,24,32,4041,43,48,4642,45,47,44,26,25,43,167,28,42,138,30,41,1117,19,24,2218,21,23,20,1,14,48,272,12,47,293,9,46,3233,35,40,3834,37,39,36,1,17,41,404,20,44,376,22,46,359,11,16,1410,13,15,12,3,38,43,195,36,45,218,33,48,2425,27,32,3026,29,31,28,1,3,8,62,5,7,49,33,25,1710,34,26,1811,35,27,19,14,22,30,3815,23,31,3916,24,32,4041,43,48,4642,45,47,44
$\mathrm{nops}\left(\mathrm{orbs}\right)$
${2}$
$\mathrm{Elements}\left(\mathrm{O2}\right)$
$\left\{{2}{\,}{4}{\,}{5}{\,}{7}{\,}{10}{\,}{12}{\,}{13}{\,}{15}{\,}{18}{\,}{20}{\,}{21}{\,}{23}{\,}{26}{\,}{28}{\,}{29}{\,}{31}{\,}{34}{\,}{36}{\,}{37}{\,}{39}{\,}{42}{\,}{44}{\,}{45}{\,}{47}\right\}$
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
GroupTheory[Elements]
GroupTheory[RubiksCubeGroup]
numelems
Perm
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