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 $\map {C_G} a$.

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 \ \ $

$\ds g^{-1} h$

$\in$

$\ds \map {C_G} a$

$g$ 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.

$\blacksquare$

1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42$. Another approach to cosets: Worked Example $2$