# Normality-small 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 article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

## Definition

### 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 $N = G$.

### In terms of the small operator

This property is obtained by applying the small operator to the property: normality
View other properties obtained by applying the small operator

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.

## Metaproperties

### Left-hereditariness

This subgroup property is left-hereditary: any subgroup of a subgroup with this property also has this property. Hence, it is also a transitive subgroup property.

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