# Wolfram Language & System 11.0 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# 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.

## DetailsDetails

• 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.

## ExamplesExamplesopen allclose all

### Basic Examples  (1)Basic Examples  (1)

Take a permutation group:

 In[1]:=

Orbit of point 3:

 In[2]:=
 Out[2]=

Point 7 belongs to the same orbit:

 In[3]:=
 Out[3]=

Point 4 belongs to a different orbit:

 In[4]:=
 Out[4]=