Orbit-stabilizer theorem

The idea is that each point of the orbit of \(x\) corresponds to a coset of the stabilizer: \(g\,\mathrm{Stab}(x)\) is matched with \(g\cdot x\). The assignment is well defined because \(g\cdot x=g'\cdot x\) is equivalent to \(g^{-1}g'\in\mathrm{Stab}(x)\), it is injective by that same equivalence, and surjective by the definition of the orbit. So the number of elements of the orbit equals the number of cosets, that is, the index.

Since the index of a subgroup is \(|G|/|H|\) by Lagrange's theorem, we get \(|\mathrm{Orb}(x)|\cdot|\mathrm{Stab}(x)|=|G|\). An immediate consequence: the size of every orbit divides \(|G|\), and if the orbit is known the stabilizer follows, and vice versa.

Guiding example: the symmetry group of the square has order \(8\) and acts on the \(4\) vertices in one orbit; so the stabilizer of a vertex has \(8/4=2\) elements, the identity and one reflection. The same count in \(S_n\) acting on \(\{1,\dots,n\}\) shows that the stabilizer of a point has \((n-1)!\) elements.

The theorem needs \(G\) finite to write quotients of orders, although the bijection between the orbit and the cosets always holds. It is later used to count orbits, to study \(p\)-groups and in the Sylow theorems.