The Orbit–Stabilizer Theorem

|G| = | orbit(x)| · | stab(x)| — where a point can go, times what holds it still.

The idea

Theorem (Orbit–stabilizer theorem).

Let a finite group $G$ act on a set $X$. For every point $x \in X$, $|G| = |\text{orbit}(x)| \cdot |\text{stab}(x)|.$

Recall the two invariants of an action: the orbit of $x$ is the set of points the group can move $x$ to, and the stabilizer of $x$ is the subgroup of elements that fix $x$. The theorem says the two sizes trade off exactly — their product is the order of the group, whichever point $x$ we pick. A point that travels far is held fixed by few elements, and a point held fixed by many elements cannot travel far.

In practice the theorem is used as division: knowing two of the three sizes gives the third.

Ways to work on it

• Walkthrough. Apply |G| = | orbit(x)| · | stab(x)| to the rotations of a cube.
• Proof. See why the theorem holds — a bijection between cosets of the stabilizer and orbit points.
• Practice. Find the missing one of group order, orbit size, and stabilizer size for a polygon's symmetries.
• Hardest. Count the rotational symmetries of a Platonic solid.

Not sure where to start? Take the ten-question placement test.