GroupOrbits

GroupOrbits[group,{p1,}]

returns the orbits of the points pi under the action of the elements of group.

GroupOrbits[group,{p1,},f]

finds the orbits under the group action given by a function f.

Details • 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 f[p,g] 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.

Examples

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Basic Examples(1)

Take a permutation group:

 In:= Orbit of point 3:

 In:= Out= Point 7 belongs to the same orbit:

 In:= Out= Point 4 belongs to a different orbit:

 In:= Out= Properties & Relations(8)

Introduced in 2010
(8.0)