| | | **The Cauchy-Frobenius Lemma 2** |

### The Cauchy-Frobenius Lemma 2

**Corollary: **
*
Let *_{G}X be a finite action and let * C* denote a
transversal of the conjugacy classes of G. Then
* | G \\X | =***(**1**)/(** | G | **)** å_{g Î C}
| C^{G}(g) | | X_{g} | =
å_{g Î C} | C_{G}(g) | ^{-1} | X_{g} | .

Here is the faster version of the
Cauchy-Frobenius Lemma.

Another formulation of the Cauchy-Frobenius Lemma makes use of the
permutation representation *g -> bar (g)* defined by the action
in question. (Actually in all our programs we apply this version of the Lemma.)
The permutation group * bar (G)* which is the image of *G*
under this representation, yields the action _{ bar (G)}X of
* bar (G)* on *X*, which has the same orbits, and so we also have:

**Corollary: **
*
If **X* denotes a finite *G*-set, then (for
*any* group *G*) the following identity holds:
* | G \\X | =***(**1**)/(** | bar (G) | **)** å_{ bar (g)
Îbar (G)} | X_{ bar (g)} | =**(**1**)/(** | bar (G) | **)**
å_{ bar (g) Îbar ( C)} | C^{ bar (G)}( bar (g))
| | X_{ bar (g)} | ,

where
* bar ( C)* denotes a transversal of the conjugacy classes of * bar (G)*.

last changed: January 19, 2005

| | | **The Cauchy-Frobenius Lemma 2** |