One-dimensional Symmetry Group - Orbits and Stabilizers

Orbits and Stabilizers

Consider a group G acting on a set X. 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:

Case that the group action is on R:

• For the trivial group, all orbits contain only one element; for a group of translations, an orbit is e.g. {..,−9,1,11,21,..}, for a reflection e.g. {2,4}, and for the symmetry group with translations and reflections, e.g., {−8,−6,2,4,12,14,22,24,..} (translation distance is 10, points of reflection are ..,−7,−2,3,8,13,18,23,..). The points within an orbit are “equivalent”. If a symmetry group applies for a pattern, then within each orbit the color is the same.

Case that the group action is on patterns:

• The orbits are sets of patterns, containing translated and/or reflected versions, “equivalent patterns”. A translation of a pattern is only equivalent if the translation distance is one of those included in the symmetry group considered, and similarly for a mirror image.

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 = yfor all g in G and all y in Y. In the example of the orbit {−8,−6,2,4,12,14,22,24,..}, {−9,−8,−6,−5,1,2,4,5,11,12,14,15,21,22,24,25,..} is invariant under G, but not fixed.

For every x in X, we define the stabilizer subgroup of x (also called the isotropy group orlittle 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 inx. 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 ofG_x. The standard quotient theorem of set theory then gives a natural bijection betweenG/G_x and Gx. Specifically, the bijection is given by hG_x |-> h · x. This result is known as the orbit-stabilizer theorem. If, in the example, we take x = 3, the orbit is {−7,3,13,23,..}, and the two groups are isomorphic with Z.

If two elements x and y belong to the same orbit, then their stabilizer subgroups, G_x andG_y, are isomorphic. More precisely: if y = g · x, thenG_y = gG_x g−1. In the example this applies e.g. for 3 and 23, both reflection points. Reflection about 23 corresponds to a translation of −20, reflection about 3, and translation of 20.