proof of counting theorem

Let $N$ be the cardinality of the set of all the couples $(g,x)$ such that $g\cdot x=x$. For each $g\in G$, there exist ${\mathrm{stab}}_g(X)$ couples with $g$ as the first element, while for each $x$, there are $|G_x|$ couples with $x$ as the second element. Hence the following equality holds:

$$N=\sum _{g\in G}{\mathrm{stab}}_g(X)=\sum _{x\in X}|G_x|.$$

From the orbit-stabilizer theorem it follows that:

$$N=|G|\sum _{x\in X}\frac{1}{|G(x)|}.$$

Since all the $x$ belonging to the same orbit $G(x)$ contribute with

$$|G(x)|\frac{1}{|G(x)|}=1$$

in the sum, then ${\sum }_{x\in X}1/|G(x)|$ precisely equals the number of distinct orbits $s$. We have therefore

$$\sum _{g\in G}{\mathrm{stab}}_g(X)=|G|s,$$

which proves the theorem.

Title	proof of counting theorem
Canonical name	ProofOfCountingTheorem
Date of creation	2013-03-22 12:47:07
Last modified on	2013-03-22 12:47:07
Owner	n3o (216)
Last modified by	n3o (216)
Numerical id	5
Author	n3o (216)
Entry type	Proof
Classification	msc 20M30