Contranormal 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]

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This is an opposite of normality

History

The notion of contranormal subgroup has been in use for quite some time, though the formal term is probably more recent.

Definition

Symbol-free definition

A subgroup of a group is contranormal if its normal closure in the group is the whole group.

Definition with symbols

To fill in.

Every maximal subgroup is either normal or contranormal.

Relation with other properties

Stronger properties

• Non-normal maximal subgroup
• Abnormal subgroup
• Strongly contranormal subgroup
• Conjugate-dense subgroup

Facts

The descendant-contranormal factorization

Every subgroup of a group can be expressed as a contranormal subgroup of a descendant subgroup. For a subgroup $H$, each term of the descending serise is the normal closuer of $H$ inside its predecessor.

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

If $G\le H\le K$ and each is contravariant in the next, then $G$ is contranormal in $K$. The proof of this follows from the fact that the normal closure of $G$ in $K$ can be obtained by first taking the normal closure of $G$ in $H$, and then again of $H$ in $K$.

Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
View other upward-closed subgroup properties

Any subgroup containing a contranormal subgroup is contranormal. This follows from the fact that the normal closure of a bigger subgroup contains the normal closure of a smaller subgroup.

Intermediate subgroup condition

Contranormality does not satisfy the intermediate subgroup condition. It seems possible that every subgroup is potentially contranormal, though a proof is not immediate.

NCI

This subgroup property is a NCI-subgroup property, i.e., it is identity-true subgroup property and further, the only normal subgroup of a group that satisfies the property is the whole group

The only normal contranormal subgroup of a group is the whole group.

Intersection-closedness

This subgroup property is not intersection-closed, viz., it is not true that an intersection of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not intersection-closed

An intersection of contranormal subgroups need not be contranormal. This follows from the fact that contranormality is an NCI-subgroup property.