Strongly 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]

This is an opposite of normality

Definition

Symbol-free definition

A subgroup of a group is termed strongly contranormal if its product with any nontrivial normal subgroup is the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed strongly contranormal in $G$ if, for any nontrivial normal subgroup $N \triangleleft G$, $HN = G$.

Relation with other properties

Related group properties

A group that possesses a strongly contranormal subgroup is termed a quasiprimitive group.

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

Any strongly contranormal subgroup of a strongly contranormal subgroup is strongly contranormal. This follows directly from the definition.

Trimness

The whole group is strongly contranormal as a subgroup of itself. In contrast, the trivial subgroup is strongly contranormal only if the group is trivial.