Subgroup with unique 2-subnormal series

Definition
A subgroup $$H$$ of a group $$G$$ is termed a subgroup with unique 2-subnormal series or subgroup whose normal closure equals its normal core of normalizer if it satisfies the following equivalent conditions:


 * 1) There is a unique subgroup $$K$$ of $$G$$ such that $$H$$ is a normal subgroup of $$K$$ and $$K$$ is a normal subgroup of $$G$$.
 * 2) The defining ingredient::normal closure $$H^G$$ of $$H$$ in $$G$$ equals the normal core of normalizer of $$H$$ in $$G$$, i.e., the defining ingredient::normal core in $$G$$ of the normalizer $$N_G(H)$$ of $$H$$ in $$G$$.