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# GroupOrbits

 GroupOrbitsreturns the orbits of the points under the action of the elements of group. GroupOrbitsfinds the orbits under the group action given by a function f.
• 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.
Take a permutation group:
Orbit of point 3:
Point 7 belongs to the same orbit:
Point 4 belongs to a different orbit:
Take a permutation group:
Orbit of point 3:
 Out[2]=
Point 7 belongs to the same orbit:
 Out[3]=
Point 4 belongs to a different orbit:
 Out[4]=
 Scope   (4)
Orbits of integer points under a permutation group:
Orbits of all points in the support of the group:
When the group only contains the identity permutation, all points are singletons:
A rank-4 symbolic tensor that is symmetric in the first and second pairs of indices:
A group of eight permutations:
Construct the orbit of tensors under the action of transposition. Only two elements are different:
 Applications   (2)
A group acts transitively in a domain if all points of the domain belong to the same orbit:
These permutations generate a transitive group:
But these do not generate a transitive group on the same domain:
The orbit of a permutation under standard permutation action is its conjugacy class:
According to the orbit-counting lemma, the number of orbits of a permutation group is equal to the average number of fixed points of its elements.
This function returns the points fixed by a permutation:
Take a group with three orbits:
Compute how many points are fixed by each element in the group:
The average is the number of orbits:
Orbits under the action of the identity group:
Group orbits of an empty list:
The lengths of the orbits are divisors of the order of the group:
If the generators have supports of very different size then usually there is one large and several small orbits:
The orbit of point 1 under a group:
Folding PermutationReplace over the group elements does not find all orbit points:
For a general expression, an orbit under Permute action is equivalent to the action of all group elements:
However, if the expression has repeated elements, then GroupOrbits will return only distinct results:
These two expressions cannot be related by a group element because they belong to different orbits:
Permutations of an alternating group cannot change signature:
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