Orbit-stabilizer theorem

(Group theory, group acting on set)

Theorem : if a group G acts on a set Ω then for each element ω in Ω -

|Orb(ω)| . |Stab(ω)| = |G|

What this theorem is saying (in a loose sense) is that for each element in Ω, G can be "factored through" by the actions which fix the element, and what's left corresponds to the elements which G takes our element to.

Note that this applies to infinite groups as well as finite ones, via the laws of arithmetic for cardinal numbers.

Proof : An application of Lagrange's theorem. Although clearly there are |G| possibilities for ωg, they are not necessarily distinct. But

ωg₁ = ωg₂
<==> ωg₁g₂^-1 = ω
<==> g₁g₂^-1 in Stab(ω)
<==> Stab(ω)g₁ = Stab(ω)g₂

So the different ωg's correspond to the different cosets of Stab(ω), and by Lagrange's theorem there are |G| / |Stab(ω) of them.

Alternative proof using coset spaces - pick an ω in Ω. Then Orb(ω) is a transitive G-space, and hence is isomorphic to (G : Stab(ω)) (as explained and proved in coset space). And by a direct application of Lagrange's theorem, |(G : Stab(ω))| = |G|/|Stab(ω)|.