The set of all conjugates of a and the set of all cosets of the Centralizer of a

[jmjlt88]

I am working on constructing a bijection between the set of all conjugates of a and the set of all cosets of the Centralizer of a. Now, I let [a]={xεG: xax^-1}. This is the set of all conjugates of a. The set {C_ax : xεG} is the set of all cosets of C_a. Hence, I want a function f: [a] -> {C_ax : x G}. I want to define f to be f(xax^-1)= (xax^-1)x. From a previous exercise, I am equipped with the fact that x^-1ax= y^-1ay if and only if C_ax= C_ay. Thus, if f(xax^-1)=f(yay^-1), then (xax^-1)x= (yay^-1)y which implies membership to both C_ax and C_ay. Hence xax^-1= yay^-1 and f is injective...

Before I go any further (i.e. prove that f is surjective and place QED at the end), is this the right idea? Or have I missed something or, perhaps, defined my function incorrectly? The fact that I am posting on here means I feel that something is amiss.

 

[micromass]

I am working on constructing a bijection between the set of all conjugates of a and the set of all cosets of the Centralizer of a. Now, I let [a]={xεG: xax^-1}. This is the set of all conjugates of a. The set {C_ax : xεG} is the set of all cosets of C_a. Hence, I want a function f: [a] -> {C_ax : x G}. I want to define f to be f(xax^-1)= (xax^-1)x.

What do you mean with thus function. We have that [itex](xax^{-1})x[/itex] is an element of G, it is not a coset of [itex]C_a[/itex]. You need f to map [itex]xax^{-1}[/itex] to a coset of [itex]C_a[/itex].

 

[jmjlt88]

Hmmm... What I am trying to do is map each element xax^-1ε[a] to C_ax for each xεG. Hence, define f(xax^-1)=C_ax... Thus, if f(xax^-1)=f(yay^-1), then C_ax= C_ay which implies x^-1ax= y^-1ay... AAAAAAAHHHHAAA! =) Better? I think it just hit me like a ton of bricks. You are the best micromass!

 

[micromass]

Be careful, you still want f to be well-defined. That is, if both [itex]xax^{-1}[/itex] and [itex]yay^{-1}[/itex] define the same element, then f of them is the same.