\begin{definition}[Definition:Conjugate (Group Theory)/Element/Also defined as]
Let $\left({G, \circ}\right)$ be a group.
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:
:$x \sim y \iff \exists a \in G: a^{-1} \circ x \circ a = y$
This is clearly equivalent to the other definition by noting that if $a \in G$ then $a^{-1} \in G$ also.
\end{definition}