Conjugates of Elements in Centralizer

Theorem
Let $$G$$ be a group.

Let $$C_G \left({a}\right)$$ be the centralizer of $$a$$ in $$G$$.

Then $$\forall g, h \in G: g a g^{-1} = h a h^{-1}$$ iff $$g$$ and $$h$$ belong to the same left coset of $$C_G \left({a}\right)$$.

Proof
The centralizer of $$a$$ in $$G$$ is defined as:

$$C_G \left({a}\right) = \left\{{x \in G: x \circ a = a \circ x}\right\}$$

Let $$g, h \in G$$. Then:

$$ $$ $$ $$

By Elements in Same Coset iff Product with Inverse in Coset, $$g$$ and $$h$$ belong to the same left coset of $$C_G \left({a}\right)$$ iff $$g^{-1} h \in C_G \left({a}\right)$$.

The result follows.