# Strongly contranormal subgroup

From Groupprops

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*

## Contents

## 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 of a group is termed **strongly contranormal** in if, for any nontrivial normal subgroup , .

## Relation with other properties

### Stronger properties

### 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.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT 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.