Intermediately normality-large subgroup

Symbol-free definition
A subgroup of a group is termed intermediately normality-large if it satisfies the following equivalent conditions:


 * 1) It is normality-large in every intermediate subgroup
 * 2) It does not occur as a retract of any subgroup properly containing it.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed intermediately normality-large in $$G$$ if it satisfies the following equivalent conditions:


 * 1) For any subgroup $$K$$ of $$G$$ containing $$H$$, and any normal subgroup $$N$$ of $$K$$ such that $$N \cap H$$ is trivial, $$N$$ must be trivial
 * 2) If $$K$$ is a subgroup of $$G$$ containing $$H$$, and $$\alpha$$ is a retraction from $$K$$ to $$H$$ (i.e. a surjective homomorphism that restricts to the identity map on $$H$$), then $$K = H$$.