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 S_{n}, 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.
As you said you studied the group action, there is a general theorem about a group action and the cardinal number of orbits.
Theorem. If a group G acts on a set S, then the cardinal number of the orbit of is the index , where is the stabilizer of x.
In your case, the group acts on S by conjugation (acting on itsef), and the orbit corresponds to the conjugacy class. So , which is the centralizer of x in S.
The cardinal number of the orbit of x = the size of conjugacy class of x = .
I think the most algebra books include the proof of the orbit and stabilizer relationship of groups.
I found this proof:
An introduction to the theory of groups - Google Books
I'm not sure how the whole coset thing gets pulled into it. I do understand that a coset is a subgroup acting on an element of G on a certain side (be it right or left).
It's not just any subgroup, its the stabilizer of x. That is really important.
You just literally build the bijection from elements in the orbits under the action into the cosets of the stablizer of x.
Well defined and injective:
Surjectivity is totally obvious by definition of orbit.