Today in class we made a table that dealt with S_8 and then we were given a handout with S_3 and S_4 and their various conjugate classes and centralizers. I was just wondering how one goes about proving the rigorous relationship between S_n where n! is the total elements and n!=centralizer*conjugate class, if there is a rigorous proof. Obviously I noticed, through Lagrange's theorem, that the order of the conjugate and centralizer both divide Sn, but is there an actual rigorous relationship? I just wanted to at least start a proof or be walk through one, kind of for kicks and giggles but also maybe for future preparation in this class.