\begin{definition}[Definition:Generated Normal Subgroup/Definition 3]
Let $G$ be a group. 
Let $S \subseteq G$ be a subset.
The '''normal subgroup generated by $S$''', denoted $\gen {S^G}$,  is the smallest normal subgroup of $G$ containing $S$:
:$\gen {S^G} = \gen {x S x^{-1}: x \in G}$
\end{definition}