Equivalence of Definitions of Conjugate of Group Element

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Theorem

The following definitions of the concept of Conjugate of Group Element are equivalent:

Definition 1

The conjugacy relation $\sim$ is defined on $G$ as:

$\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x = y \circ a$

Definition 2

The conjugacy relation $\sim$ is defined on $G$ as:

$\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x \circ a^{-1} = y$


Also defined as

Some sources define the conjugate of $x$ by $a$ in $G$ as:

$x \sim y \iff \exists a \in G: x \circ a = a \circ y$

or:

$x \sim y \iff \exists a \in G: a^{-1} \circ x \circ a = y$


Proof

\(\ds a \circ x\) \(=\) \(\ds y \circ a\)
\(\ds \leadstoandfrom \ \ \) \(\ds a \circ x \circ a^{-1}\) \(=\) \(\ds y\)


\(\ds x \circ a\) \(=\) \(\ds a \circ y\)
\(\ds \leadstoandfrom \ \ \) \(\ds a^{-1} \circ x \circ a\) \(=\) \(\ds y\)


\(\ds \exists b \in G: \, \) \(\ds x \circ b\) \(=\) \(\ds b \circ y\)
\(\ds \leadstoandfrom \ \ \) \(\ds b^{-1} \circ x \circ b \circ b^{-1}\) \(=\) \(\ds b^{-1} \circ b \circ y \circ b^{-1}\)
\(\ds \leadstoandfrom \ \ \) \(\ds b^{-1} \circ x\) \(=\) \(\ds y \circ b^{-1}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \exists a \in G: \, \) \(\ds a \circ x\) \(=\) \(\ds y \circ a\) setting $a := b^{-1}$

$\blacksquare$