top half. Group action Consider D3 in the geometrical way, as a symmetry group of isometries of the plane, and consider the corresponding group action on a set of 30 evenly spaced points on a circle, numbered 0 to 29, with 0 at one of the reflexion axes. This section illustrates group action concepts for this case. The action of G on X is called transitive if for any two x, y in X there exists a g in G such that ; this is not the case faithful (or effective) if for any two different g, h in G there exists an x in X such that ; this is the case, because, except for the identity, symmetry groups do not contain elements that "do nothing" free if for any two different g, h in G and all x in X we have ; this is not the case because there are reflections Orbits and stabilizers The orbit of a point x in X is the set of elements of X to which x can be moved by the elements of G. The orbit of x is denoted by Gx: The orbits are and The points within an orbit are "equivalent". If a symmetry group applies for a pattern, then within each orbit the color is the same. The set of all orbits of X under the action of G is written as . If Y is a subset of X, we write GY for the set We call the subset Y invariant under G if (which is equivalent to . In that case, G also operates on Y. The subset Y is called fixed under G if for all g in G and all y in Y. The union of e.g. two orbits is invariant under G, but not fixed. For every x in X, we define the stabilizer subgroup of x (also called the isotropy group or little group) as the set of all elements in G that fix x: If x is a reflection point , its stabilizer is the group of order two containing the identity and the reflection in x. In other cases the stabilizer is the trivial group. For a fixed x in X, consider the map from G to X given by . The image of this map is the orbit of x and the coimage is the set of all left