I am working on constructing a bijection between the set of all conjugates of(adsbygoogle = window.adsbygoogle || []).push({}); aand the set of all cosets of the Centralizer ofa. Now, I let [a]={xεG: xax^{-1}}. This is the set of all conjugates of a. The set {C_{a}x : xεG} is the set of all cosets of C_{a}. Hence, I want a function f: [a] -> {C_{a}x : x G}. I want to definefto bef(xax. From a previous exercise, I am equipped with the fact that^{-1})= (xax^{-1})xxif and only if C^{-1}ax= y^{-1}ay_{a}x= C_{a}y. Thus, iff(xax, then^{-1})=f(yay^{-1})(xaxwhich implies membership to both C^{-1})x= (yay^{-1})y_{a}x and C_{a}y. Hencexaxand^{-1}= yay^{-1}fis injective....

Before I go any further (i.e. prove thatfis 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.

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# Homework Help: The set of all conjugates of a and the set of all cosets of the Centralizer of a

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