NE-subgroup

Origin
The notion of NE-subgroup weas used by Yangming Li in his paper in the Journal of Group Theory.

Symbol-free definition
A subgroup of a group is termed a NE-subgroup if it equals the intersection of its normalizer and normal closure.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a NE-subgroup if $$H = N_G(H)$$ &cap; $$H^G$$.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::Self-normalizing subgroup

Weaker properties

 * Stronger than::Weakly normal subgroup
 * Stronger than::Intermediately subnormal-to-normal subgroup
 * Stronger than::Subgroup with self-normalizing normalizer

Metaproperties
Since the normalizer inside an intermediate subgroup is contained inside the whole normalizer, and since the normal closure inside an intermediate subgroup is also contained inside the whole normal subgroup, we have the following: If $$H$$ is a NE-subgroup of $$G$$ and $$K$$ is an intermediate subgroup containing $$H$$, then $$H$$ is a NE-subgroup of $$K$$. IN other words, $$H$$ satisfies the intermediate subgroup condition.