# Subgroup whose normal closure is fully invariant

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]

## Contents

## Definition

A subgroup of a group is termed a **subgroup whose normal closure is fully invariant** if , its normal closure in the whole group, is a fully invariant subgroup of .

## Examples

### Extreme examples

- The trivial subgroup satisfies this property in every group.
- Every group satisfies this property in itself.

### Important classes of examples

- All subgroups inside simple groups, as well as all contranormal subgroups, satisfy this property.
- Sylow subgroups, as well as subgroups arising as a join of Sylow subgroups (and in particular Hall subgroups) satisfy this property.

## Relation with other properties

### Stronger properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

closure-characteristic subgroup | normal closure is a characteristic subgroup | fully invariant implies characteristic | follows by combining characteristic not implies fully invariant and the fact that, since characteristic implies normal, a characteristic subgroup is its own normal closure | |FULL LIST, MORE INFO |

subgroup whose characteristic closure is fully invariant | characteristic closure is a fully invariant subgroup | if the normal closure is fully invariant, it is also characteristic (because fully invariant implies characteristic), and since characteristic implies normal, it is the characteristic closure | proper nontrivial subgroup in a characteristically simple group, e.g., Z2 in V4 | |FULL LIST, MORE INFO |