### Cosets

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/G*_{x} :gx -> gG_{x} is a
bijection.

Proof: It is clear that, for a given *x ÎX*,
the following chain of equivalences holds:

*gx=g'x iff g*^{-1}g' ÎG_{x} iff g'G_{x}=gG_{x}.

Reading it from left to right we see that *gx -> gG*_{x}
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 | / | G*_{x} | .

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:
* | C*^{G}(g) | = | G | / | C_{G}(g) | ,
and | [~U] | =
| G | / | N_{G}(U) | .

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001