Normality-large subgroup

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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.

Relation with other properties

Stronger properties

• Normality-large normal subgroup
• Self-centralizing normal subgroup::For full proof, refer: Self-centralizing and normal implies normality-large

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.
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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.