proof of counting theorem

Let N be the cardinality of the set of all the couples (g,x) such that gx=x. For each gG, there exist stabg(X) couples with g as the first element, while for each x, there are |Gx| couples with x as the second element. Hence the following equality holds:


From the orbit-stabilizer theorem it follows that:


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


in the sum, then xX1/|G(x)| precisely equals the number of distinct orbits s. We have therefore


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