\begin{definition}[Definition:Normal Closure]
Let $\left({G, \circ}\right)$ be a group.
Let $S$ be a subset of $G$.
The '''normal closure of $S$ in $G$''' is the smallest normal subgroup of $G$ that contains $S$ as a subset.
The normal closure is equal to the conjugate closure $\left\langle S^G \right\rangle := \left\langle \left\{{x \circ s \circ x^{-1}: x \in G, s \in S}\right\}\right\rangle$
which is the subgroup generated by the set of all elements of $G$ that are conjugate to an element of $S$.
\end{definition}