Conjugates of Elements in Centralizer

Theorem
Let $G$ be a group.

Let $\map {C_G} a$ be the centralizer of $a$ in $G$.

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

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


 * $\map {C_G} a = \set {x \in G: x \circ a = a \circ x}$

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 $\map {C_G} a$ $g^{-1} h \in \map {C_G} a$.

The result follows.