| | | **The Cauchy-Frobenius Lemma** |

### The Cauchy-Frobenius Lemma

The previous result is very important, it is
essential in the proof of
the following counting lemma which, together with later refinements,
forms *the basic tool of the theory of enumeration under finite group
action*:
*
***The Lemma of Cauchy-Frobenius**
The number of orbits of a finite group *G* acting on a finite set *X*
is equal to the average
number of fixed points:
* | G \\X | =***(**1**)/(** | G | **)** å_{g ÎG}
| X_{g} | .

Proof:

* å*_{g ÎG} | X_{g} | = å_{g} å_{x ÎXg} 1
= å_{x} å_{g ÎGx} 1 = å_{x} | G_{x} | ,

which is, by the index formula, equal to
* | G | å*_{x}
| G(x) | ^{-1} = | G | · | G \\X | .

Now you can try to make some calculations using the
Cauchy-Frobenius Lemma.

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

| | | **The Cauchy-Frobenius Lemma** |