GroupOrbits

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:

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: