The Cauchy-Frobenius LemmaActions of groupsExamplesCosets


Returning to the general case we first state the main (and obvious) properties of the stabilizers of elements belonging to the same orbit: But the crucial point is the following natural bijection between the orbit of x and the set of left cosets of its stabilizer:
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) | .,
last changed: August 28, 2001

The Cauchy-Frobenius LemmaActions of groupsExamplesCosets