# 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

open allclose all

## Basic Examples(1)

Take a permutation group:

Orbit of point 3:

Point 7 belongs to the same orbit:

Point 4 belongs to a different orbit:

## 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(3)

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:

Check whether a simple and connected graph is distance-transitive:

These graphs are distance-transitive:

These graphs are not distance-transitive:

## Properties & Relations(8)

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: