\begin{definition}[Definition:Conjugate (Group Theory)/Element/Definition 2]
Let $\struct {G, \circ}$ be a group.
The  '''conjugacy relation''' $\sim$ is defined on $G$ as:
:$\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x \circ a^{-1} = y$
This can be voiced as:
:'''$x$ is the conjugate of $y$ (by $a$ in $G$)'''
or:
:'''$x$ is conjugate to $y$ (by $a$ in $G$)'''
\end{definition}