Definition:Conjugate (Group Theory)/Element/Also defined as

Definition
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.

Also see

 * Equivalence of Definitions of Conjugate of Group Element