The Cauchy-Frobenius Lemma 2

### The Cauchy-Frobenius Lemma 2

Corollary: Let GX be a finite action and let C denote a transversal of the conjugacy classes of G. Then
| G \\X | =(1)/( | G | ) åg Î C | CG(g) | | Xg | = åg Î C | CG(g) | -1 | Xg | .

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).

harald.fripertinger@kfunigraz.ac.at,
last changed: August 28, 2001

 The Cauchy-Frobenius Lemma 2