Orbit-stabilizer theorem
From AoPSWiki
The orbit-stabilizer theorem is a combinatorial result in group theory.
Let 
be a group acting on a set 
. For any 
, let 
denote the stabilizer of 
, and let 
denote the orbit of . The orbit-stabilizer theorem states that
Proof. Without loss of generality, let operate on from the right. We note that if 
are elements of such that 
, then 
. Hence for any 
, the set of elements 
of for which 
constitute a unique left coset modulo . Thus
The result then follows from Lagrange's Theorem. 
