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 Left Coset iff Product with Inverse in Subgroup, $g$ 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.