Orbit-stabilizer theorem
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
![\[\lvert G \rvert = \lvert \text{orb}(i) \rvert \cdot \lvert \text{stab}(i) \rvert .\]](http://latex.artofproblemsolving.com/4/f/d/4fd05faa02a0317f2d3ff70c1b49a4c2b97f22ff.png)
Proof. Without loss of generality, let operate on from the left. 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
![\[\lvert \text{orb}(i) \rvert = \lvert G/\text{stab}(i) \rvert.\]](http://latex.artofproblemsolving.com/2/b/d/2bdf56fb75cf03827521db5982b2acb72f3f7fca.png)
The result then follows from Lagrange's Theorem. 
