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]

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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

Stronger properties

• Core-free maximal subgroup

Weaker properties

• Contranormal subgroup
• Core-free subgroup (if it is not the whole group)

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.
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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.