Strongly contranormal subgroup

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 $$N \triangleleft G$$, $$HN = G$$.

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
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.