Let \(G\) be a group and \(H\leq G\) a subgroup. The normal closure of \(H\) in \(G\) is the subgroup generated by the conjugates of \(H\): \[ H^G \coloneqq \gens{ \Union_{g\in G} gHg^{-1} }. \]
Let \(G\) be a group and \(H\leq G\). Then \(H^G\) is the smallest normal subgroup of \(G\) containing \(H\): \[ H^G = \bigcap \theset{N \st H \leq N \normal G }. \]