GroupOrbits

GroupOrbits[group,{p1,}]
returns the orbits of the points under the action of the elements of group.

GroupOrbits[group,{p1,},f]
finds the orbits under the group action given by a function f.

DetailsDetails

  • Two points belong to the same orbit under group if there is an element g in group such that the image of one point under g is the other point.
  • If a point p is fixed by all elements in group then it forms an orbit {p}.
  • GroupOrbits[group] gives all orbits in the natural domain of action of group.
  • Orbits are given as sorted lists.
  • Evaluation of for an action function f, a point p and a permutation g of the given group, is assumed to return another point p'.
  • For permutation groups, the default group action is taken to be PermutationReplace.

ExamplesExamplesopen allclose all

Basic Examples  (1)Basic Examples  (1)

Take a permutation group:

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Orbit of point 3:

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Point 7 belongs to the same orbit:

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Point 4 belongs to a different orbit:

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Introduced in 2010
(8.0)