Normal closure of a subgroup

If $H$ is a {{KNOWL('group.subgroup', 'subgroup')}} of a group $G$, the **normal closure** of $H$ is the smallest {{KNOWL('group.subgroup.normal', 'normal subgroup')}} of $G$ containing $H$. Alternatively, it is the subgroup generated by the set $$ \bigcup_{g\in G} gHg^{-1}. $$