# Normal closure-closed subgroup property

From Groupprops

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

## Contents

## Definition

### Symbol-free definition

A subgroup property is said to be **normal closure-closed** if whenever a subgroup has the property in the whole group, its normal closure also has the property.

### Definition with symbols

A subgroup property is said to be **normal core-closed** if whenever satisfies property in , the normal core also satisfies in .

## Relation with other metaproperties

### Stronger metaproperties

- Join-closed subgroup property
- Finite-join-closed subgroup property when we are guaranteed that there are only finitely many conjugates
- Subgroup properties stronger than normality
- Identity-true subgroup properties stronger than contranormality

### Related metaproperties

- Normalizer-closed subgroup property
- Normal core-closed subgroup property
- Characteristic closure-closed subgroup property

### Opposite metaproperties

- NCT-subgroup property that is
*not*stronger than the property of being the trivial subgroup:`For full proof, refer: NCT and not trivial implies not normal closure-closed` - NCI-subgroup property that is
*not*stronger than contranormality:`For full proof, refer: NCT and not contranormal implies not normal closure-closed`

## Subgroup properties that are normal closure-closed

### Join-closed subgroup properties

One way of establishing that a subgroup property is normal closure-closed is to show that the join of any family of subgroups having the property, also has the property.

### Subgroup properties stronger than normality

If a subgroup property is stronger than normality, then the normal closure of any subgroup having the property is itself, and hence whenever a subgroup has the property, so does its normal closure.