Group actions and orbit–stabilizer

A left action of G on a set X assigns an element g·x of X to each pair g,x. The identity must leave every x fixed, and applying h followed by g must equal applying gh. Each g therefore acts bijectively, because its inverse undoes the action. Sending g to this permutation of X defines a homomorphism into the symmetric group of X. An action is faithful when only the identity acts as the identity permutation on every point.

Fix a point x. Its orbit contains every point reachable as g·x, and its stabilizer contains the operations that leave this particular point unchanged. The stabilizer is a subgroup: identity fixes x, products of fixing operations fix x, and inverses do too. The action kernel is the intersection of all point stabilizers. An operation can fix one point while moving another, so belonging to one stabilizer does not place it in the kernel. Orbits partition X because reachability is reflexive, symmetric, and transitive.

The map from left cosets of G_x to the orbit of x sends gG_x to g·x. It is well defined because multiplying g on the right by something that fixes x cannot change the result. Conversely, if g·x=h·x, then h inverse g fixes x, so g and h lie in the same coset. Every orbit point is reached by definition. This constructs a bijection, proving orbit–stabilizer. The stabilizer need not be normal: the cosets are being counted as a set, not multiplied as a quotient group.

For finite G, the result says that the orbit size times stabilizer size equals |G|. In the vectors lab, hold magnitude positive and vary angle through quarter turns. This illustrates the orbit of a nonzero planar vector under the four rotations of a square. The lab permits continuous angles and displays one vector with its projections; it does not enumerate a discrete orbit or compute stabilizers. A zero vector is special: every rotation fixes it, so its orbit has one element rather than four.

Sometimes the goal is to count inequivalent objects, not the size of one orbit. Burnside’s lemma averages the number of fixed objects over the group. To see why, count pairs (g,x) satisfying g·x=x. Counting first by g gives the sum of fixed-point counts; counting first by x gives the sum of stabilizer sizes. Within any one orbit, all stabilizers have equal size and orbit–stabilizer makes their total contribution exactly |G|. Dividing by |G| therefore counts orbits.

Specify the allowed symmetries before counting. A necklace considered equivalent under rotations has a different acting group from one allowing reflections as well. Colors are also part of the acted-on set: if swapping color names is not declared a symmetry, it must not be silently included. In small problems, list fixed configurations for each operation and verify that the average is an integer. This is more reliable than dividing the total number of configurations by |G|, which works only when every stabilizer is trivial.