Normality-small subgroup

Symbol-free definition
A subgroup of a group is termed normality-small if the only normal subgroup with which its product is the whole group, is in fact the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed normality-small if whenever $$N \triangleleft G$$ is such that $$HN = G$$, we have $$N = G$$.

The small operator takes as input a subgroup property and outputs the property of being a subgroup whose join with every proper subgroup having this property is proper. The subgroup property of being normality-small is obtained by applying the small operator to the subgroup property of normality.

Related properties

 * Normality-large subgroup
 * Strongly contranormal subgroup
 * Frattini-embedded normal subgroup

Metaproperties
Any subgroup sitting inside a normality-small subgroup is normality-small. This follows directly from the definition.