Normality-large subgroup

Symbol-free definition
A subgroup in a group is termed normality-large if it satisfies the following condition: Any normal subgroup that intersects it trivially, must also be trivial.

Definition with symbols
A subgroup $$H$$ in a group $$G$$ is termed normality-large if, whenever $$N$$ is a normal subgroup of $$G$$ such that $$N \cap H$$ is trivial, $$N$$ is itself trivial.

Formalisms
The property of being normality-large is obtained by applying the large operator to the subgroup property of normality.

Stronger properties

 * Weaker than::Normality-large normal subgroup
 * Weaker than::Self-centralizing normal subgroup::

Metaproperties
The property of being normality-large is a transitive subgroup property. In other words, any normality-large subgroup of a normality-large subgroup is normality-large. This follows from the theory of the large operator and the fact that normality satisfies the transfer condition.

Intermediate subgroup condition
The property of being normality-large does not in general satisfy the intermediate subgroup condition.

Trimness
The property of being normality-large is identity-true, that is, it is satisfied by the whole group as a subgroup of itself. However, it is not trivially true.

Intersection-closedness
An intersection of normality-large subgroups is not in general normality-large. One example of this is a proper nontrivial subgroup in a simple group. This is normality-large, but the intersection of all its conjugates is trivial, and that is not normality-large.

Intermediately operator
An intermediately normality-large subgroup is a subgroup that is normality-large in every intermediate subgroup. This is equivalent to the condition that this subgroup does not occur as a retract of any subgroup properly containing it.

Intersection transiter
While an intersection of two normality-large subgroups is not in general normality-large, the intersection of a normality-large subgroup and a normality-large normal subgroup is always normality-large. Thus, the property of being a normality-large normal subgroup is stronger than the intersection transiter of the property of being a normality-large subgroup.