Definition:Conjugate (Group Theory)

Let $$\left({G, \circ}\right)$$ be a group.

Then an element $$x \in G$$ is conjugate to an element $$y \in G$$ iff:

$$\exists a \in G: x \circ a = a \circ y$$

Alternatively, we can say that "$$x$$ is the conjugate of $$y$$ by $$a$$".

This relation is called conjugacy, and we write $$x \sim y$$ for "$$x$$ is a conjugate of $$y$$".