Definition:Conjugate (Group Theory)/Element/Definition 2

Definition
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 $G$)

Also see

 * Conjugacy is Equivalence Relation
 * Definition:Conjugacy Class