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

\(\displaystyle a \circ x\) \(=\) \(\displaystyle y \circ a\)
\(\displaystyle \iff \ \ \) \(\displaystyle a \circ x \circ a^{-1}\) \(=\) \(\displaystyle y\)


\(\displaystyle x \circ a\) \(=\) \(\displaystyle a \circ y\)
\(\displaystyle \iff \ \ \) \(\displaystyle a^{-1} \circ x \circ a\) \(=\) \(\displaystyle y\)


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

$\blacksquare$