$$\begin{array}{cl} \sum_{\alpha\in \Omega}|G_{\alpha}| & =\sum_{\alpha\in \Omega}\frac{|G|}{|G\cdot\alpha|} \\ & =|G|\sum_{\alpha\in \Omega}\frac{1}{|G\cdot\alpha|} \\ & =|G|\sum_{A\in X/G}\sum_{\alpha\in A}\frac{1}{|A|} \\ & =|G|\sum_{A\in X/G}1 \\ &=|G||X/G|.\end{array}$$
But I don't understand why can I do $$|G|\sum_{\alpha\in \Omega}\frac{1}{|G\cdot\alpha|}=|G|\sum_{A\in X/G}\sum_{\alpha\in A}\frac{1}{|A|}~?$$
Orbits partition $G$-sets. Thus summing over all of $\Omega$ is the same as summing the particular summations over each orbit. And if $A$ is an orbit then $|A|=|G\alpha|$ for each $\alpha\in A$; it's the same orbit no matter which representative $\alpha$ you pick. – anon Oct 20 '12 at 16:44