orbit-stabilizer theorem

There is a clean trade-off in any group action: the more symmetry pins a point down, the fewer places that point can travel; the more freely a point moves, the less holds it fixed. The orbit-stabilizer theorem makes this trade-off exact — orbit size and stabilizer size multiply to give the size of the group.

Precisely, for an action of a group G on a set X and any point x, there is a bijection between the orbit G·x and the set of left cosets G/Stab(x), given by g·x <-> g·Stab(x). Hence |G·x| = [G : Stab(x)], the index of the stabilizer. When G is finite this reads |G| = |G·x| · |Stab(x)|, so both the orbit length and the stabilizer order divide |G|.

This single identity underlies an enormous amount of finite group theory. It is the engine behind Lagrange's theorem (take G acting on cosets), the class equation (take G acting on itself by conjugation), and Burnside's counting lemma. A practical reading: to find an orbit's size, you do not need to list it — just compute how many group elements fix one point and divide |G| by that number.