normal closure

Let \(G\) be a group and \(S \subseteq G\) a subset.
Then the normal closure \(\mathrm{ncl}(S)\) is defined as smallest normal subgroup with \(S \subseteq \mathrm{ncl}(S)\)

\begin{align*} \bigcap_{ \substack{ N \text{ normal subgroup} \\ S \subseteq N }} N \end{align*}

see: Schnitt von Normalteilern als Normalteiler