Lemma: The mapping G(x) -> G/Gx :gx -> gGx is a bijection.
Proof: It is clear that, for a given x ÎX, the following chain of equivalences holds:
gx=g'x iff g-1g' ÎGx iff g'Gx=gGx.Reading it from left to right we see that gx -> gGx defines a mapping, reading it from right to left we obtain that it is injective. Furthermore it is obvious that this mapping is also surjective.
This result shows in particular that the length of the orbit is the index of the stabilizer, so that we obtain
Corollary: If G is a finite group acting on the set X, then for each x ÎX we have| G(x) | = | G | / | Gx | .
An application to the examples given above yields:
Corollary: If G is finite, g ÎG, and U £G, then the orders of the conjugacy classes of elements and of subgroups satisfy the following equations:| CG(g) | = | G | / | CG(g) | , and | [~U] | = | G | / | NG(U) | .