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 allBasic Examples (1)
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:
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:
Text
Wolfram Research (2010), GroupOrbits, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupOrbits.html.
CMS
Wolfram Language. 2010. "GroupOrbits." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GroupOrbits.html.
APA
Wolfram Language. (2010). GroupOrbits. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GroupOrbits.html