Conjugates of Elements in Centralizer

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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 $g, h \in G$.


\(\ds g a g^{-1}\) \(=\) \(\ds h a h^{-1}\)
\(\ds \leadstoandfrom \ \ \) \(\ds g^{-1} \paren {g a g^{-1} } h\) \(=\) \(\ds g^{-1} \paren {h a h^{-1} } h\)
\(\ds \leadstoandfrom \ \ \) \(\ds a g^{-1} h\) \(=\) \(\ds g^{-1} h a\)
\(\ds \leadstoandfrom \ \ \) \(\ds g^{-1} h\) \(\in\) \(\ds \map {C_G} a\) Definition of Centralizer of Group Element

By Elements in Same Left Coset iff Product with Inverse in Subgroup:

$g$ and $h$ belong to the same left coset of $\map {C_G} a$ if and only if $g^{-1} h \in \map {C_G} a$.

The result follows.