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]
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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 in a group is termed normality-large if, whenever is a normal subgroup of such that is trivial, is itself trivial.
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
Relation with other properties
- Normality-large normal subgroup
- Self-centralizing normal subgroup::For full proof, refer: Self-centralizing and normal implies normality-large
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.
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
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.
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.