Subgroup whose normalizer equals its normal closure

Definition
A subgroup $$H$$ of a group $$G$$ is termed a subgroup whose normalizer equals its normal closure if it satisfies the following equivalent conditions:


 * 1) The defining ingredient::normalizer $$N_G(H)$$ of $$H$$ in $$G$$ equals the defining ingredient::normal closure $$H^G$$ of $$H$$ in $$G$$.
 * 2) It is both a 2-hypernormalized subgroup (its normalizer is normal) and a subgroup with unique 2-subnormal series (it has a unique subnormal series of length two).