\begin{definition}[Definition:Conjugate (Group Theory)/Subset]
Let $\struct {G, \circ}$ be a group.
Let $S \subseteq G, a \in G$.
Then the '''$G$-conjugate of $S$ by $a$''' is:
:$S^a := \set {y \in G: \exists x \in S: y = a \circ x \circ a^{-1} } = a \circ S \circ a^{-1}$
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, we usually just refer to the '''conjugate of $S$ by $a$'''.
\end{definition}