Let $G$ be a group and $\sigma \in G$ some element. I want to prove that there is a injuctive homomorphism between the left cosets of the centralizer $C_G(\sigma)$ and between the Conjugacy classes of $\sigma$, that are $\{h\sigma h^{-1}\mid h\in G\}$.
I defined $f(hC_G(\sigma)) = h\sigma h^{-1}$.
I think it is a homomorphism but I couldn't manage to prove it. I am not even sure what is the operation that should be here. Is it multiplication, composition or Conjugacy?
If it is a homomorphism I did however managed to prove it is injuctive. So with this part I am alright.
Help would be appreciated