# Conjugates of Elements in Centralizer

## Theorem

Let $G$ be a group.

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

Then $\forall g, h \in G: g a g^{-1} = h a h^{-1}$ if and only if $g$ 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:

 $\ds g a g^{-1}$ $=$ $\ds h a h^{-1}$ $\ds \leadstoandfrom \ \$ $\ds g^{-1} \paren {g a g^{-1} } h$ $=$ $\ds g^{-1} \paren {h a h^{-1} } h$ $\ds \leadstoandfrom \ \$ $\ds a g^{-1} h$ $=$ $\ds g^{-1} h a$ $\ds \leadstoandfrom \ \$ $\ds g^{-1} h$ $\in$ $\ds \map {C_G} a$ Definition of Centralizer of Group Element
$g$ and $h$ belong to the same left coset of $\map {C_G} a$ if and only if $g^{-1} h \in \map {C_G} a$.

The result follows.

$\blacksquare$