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At this time, I am reading the following theorem.

Let $G$ be a group acting transitively on a set $\Omega$. Then \begin{equation} |G|=\sum_{g\in G}\chi(g) \tag{$\clubsuit$} \end{equation} wherein $\chi(g)=|\{\alpha \in \Omega :\alpha^g=\alpha\}|$.

It seems to me that the equation $(\clubsuit)$ is very similar to the class equation for a finite group $G$, \[ |G|=|Z(G)|+\sum_{i=1}^{k}\frac{|G|}{|C_G(x_i)|}.\] Are these equations connected to each other? Thanks.

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  • $\begingroup$ I hope my edits are alright. Note that your ($\clubsuit$) is a special case [here $|G\backslash\Omega| = 1$] of Burnside's lemma. $\endgroup$ – Dylan Moreland Jun 21 '12 at 20:38
  • $\begingroup$ @DylanMoreland: Thanks for edit and your time. I am thinking about the proper action I should take. Thanks again. $\endgroup$ – Mikasa Jun 21 '12 at 20:47
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The class equation is really a special case of the fact that if $G$ acts (not necessarily transitively) on a set $\Omega,$ then $|\Omega|$ is the sum of the size of the orbits. In the class equation, $G$ acts on itself by conjugation, the orbits are the conjugacy classes, and the size of the conjugacy class of $x$ is $[G:C_{G}(x)].$ The euation as you have written it is often known as the modified class equation. In that, all the conjugacy classes of size $1$ have been collected together, since the element $x$ is in a conjugacy class of size $1$ if and only if $x \in Z(G).$ There is a connection in as much as when a finite group $G$ acts transitively on a set $\Omega,$ then all point stabilizers have the same size$|G_{\alpha}|$ for any $\alpha \in \Omega$ so we have $|\Omega| = [G:G_{\alpha}].$ Thus $\sum_{ \alpha \in \Omega }|G_{\alpha}| = |G|.$ But that sum is the number of order pairs $(\alpha, g) \in \Omega \times G$ such that $\alpha.g = \alpha.$ If we sum this over $g \in G$ first, it's the same as $\sum_{g \in G} \# (\alpha \in \Omega : \alpha.g = \alpha).$

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  • $\begingroup$ Conjugation was what I was looking for. I got it all.Thanks. $\endgroup$ – Mikasa Jun 21 '12 at 21:03

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