Dihedral Group of Order 6 - Orbits and Stabilizers

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 {0,10,20}, {1,9,11,19,21,29}, {2,8,12,18,22,28}, {3,7,13,17,23,27}, {4,6,14,16,24,26}, and {5,15,25}. 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 X / G.

If Y is a subset of X, we write GY for the set { g · y : y Y and g G }. We call the subset Y invariant under G if GY = Y (which is equivalent to GY ⊆ Y). In that case, G also operates on Y. The subset Y is called fixed under G if g · y = y 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 (0, 5, 10, 15, 20, or 25), 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 g |-> g · x. The image of this map is the orbit of x and the coimage is the set of all left cosets of G_x. The standard quotient theorem of set theory then gives a natural bijection between G/G_x and Gx. Specifically, the bijection is given by hG_x |-> h · x. This result is known as the orbit-stabilizer theorem. In the two cases of a small orbit, the stabilizer is non-trivial.

If two elements x and y belong to the same orbit, then their stabilizer subgroups, G_x and G_y, are isomorphic. More precisely: if y = g · x, then G_y = gG_x g−1. In the example this applies e.g. for 5 and 25, both reflection points. Reflection about 25 corresponds to a rotation of 10, reflection about 5, and rotation of -10.

A result closely related to the orbit-stabilizer theorem is Burnside's lemma:

where Xg is the set of points fixed by g. I.e., the number of orbits is equal to the average number of points fixed per group element.

For the identity all 30 points are fixed, for the two rotations none, and for the three reflections two each: {0,15}, {5,20}, and {10, 25}. Thus the average is six, the number of orbits.