# Normality-large 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]

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

This subgroup property is obtained by applying the large operator to the subgroup property:normal subgroup

## Definition

### 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

### In terms of the large operator

This property is obtained by applying the large operator to the property: normal subgroup
View other properties obtained by applying the large operator

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

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

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.

For full proof, refer: Normality-largeness is transitive

### 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.

## Effect of property operators

### The intermediately operator

Applying the intermediately operator to this property gives: intermediately normality-large subgroup

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.