# What about the Cauchy-Frobenius-orbit-counting formula

I know the proposition that says:

Let $\lambda$ be a homomorphism from a finite group $G$ into $\mathbb{C}^{\times}$. Suppose that $G$ acts on some finite set $\Omega$ and let $M$ be the number of $\lambda$-good orbits for this action. Then $$M=\frac{1}{|G|}\sum_{g\in G}\lambda(g)\pi(g),$$ where $\pi$ is the permutation character associated with the action.

Using this, I need to prove:

Let $G$ be a finite group acting on a finite set $\Omega$, and let $\lambda$ be an arbitrary homomorphism from $G$ into $\mathbb{C}^{\times}$. Then for each positive integer $a$: $$\sum_{g\in G}\lambda(g)a^{c(g)} \equiv 0 \pmod{|G|},$$ where $c(g)$ is the number of orbits of $\langle g \rangle$ on $\Omega$.

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What does $\lambda$-good mean, by the by? – anon Oct 20 '12 at 0:44
if $\mathcal{O}$ is a $G$-orbit on $\Omega$, then $\mathcal{O}$ is a $\lambda-good$ if the stabilizer in $G$ of every point in $\mathcal{O}$ is contained in $ker(\lambda)$ – Elmo goya Oct 20 '12 at 0:50

## 1 Answer

See "Corollary F" in Generalizations of Fermat's Little Theorem via Group Theory by Pournaki and Isaacs.

If $G: \Omega \to \Omega$ is a group action. This can be extended to a permutation action on the power set $\Omega^A = \{ f: \Omega \to A\}$. The number of fixed points of this permutation is your character $\chi(g)$.

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