# NE-subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

## Origin

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

## Definition

### 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 $H = N_G(H)$ $H^G$.

## Metaproperties

### Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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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.