Definition:Conjugate (Group Theory)

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

Conjugate of an Element
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 $$y$$".

Alternative definitions
There are several ways of defining conjugacy of elements, all of them trivially equivalent.

$$x$$ is a conjugate of $$y$$ iff:
 * $$\exists a \in G: a \circ x = y \circ a$$;
 * $$\exists a \in G: a \circ x \circ a^{-1} = y$$;
 * $$\exists a \in G: a^{-1} \circ x \circ a = y$$.

Conjugate of a Set
Let $$S \subseteq G, a \in G$$.

Then the $$G$$-conjugate of $$S$$ by $$a$$ is:

$$\mathbf {Define:} \ S^a \ \stackrel {\mathbf {def}} {=\!=} \ \left\{{y \in G: \exists x \in S: y = a \circ x \circ a^{-1}}\right\} = a \circ S \circ a^{-1}$$

That is, $$S^a$$ is the set of all elements of $$G$$ that are the conjugates of elements of $$S$$ by $$a$$.

When $$G$$ is the only group under consideration (as is usual), we usually just refer to the conjugate of $$S$$ by $$a$$.