# Normal subgroup of characteristic subgroup

From Groupprops

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: normal subgroup and characteristic subgroup

View other such compositions|View all subgroup properties

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed a **normal subgroup of characteristic subgroup** if it satisfies the following equivalent conditions:

- It is a normal subgroup of a characteristic subgroup of the group.
- It is normal inside its characteristic closure in the group.
- Its characteristic closure is contained in its normalizer.
- It is contained in the characteristic core of its normalizer.

### Definition with symbols

A subgroup of a group is termed a **normal subgroup of characteristic subgroup** if it satisfies the following equivalent conditions:

- There exists a characteristic subgroup of such that is a normal subgroup of .
- is a normal subgroup inside the characteristic closure of in .
- The characteristic closure of in is contained in the normalizer .
- is contained in the characteristic core of in .

## Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this propertyVIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

## Relation with other properties

### Stronger properties

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

normal subgroup | |FULL LIST, MORE INFO | |||

characteristic subgroup | Direct factor of characteristic subgroup, Left-transitively 2-subnormal subgroup|FULL LIST, MORE INFO | |||

direct factor of characteristic subgroup | |FULL LIST, MORE INFO | |||

left-transitively 2-subnormal subgroup | Left-transitively 2-subnormal implies normal of characteristic | |FULL LIST, MORE INFO |

### Weaker properties

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

automorph-permutable subgroup | |FULL LIST, MORE INFO | |||

2-subnormal subgroup | |FULL LIST, MORE INFO | |||

3-subnormal subgroup | 2-subnormal subgroup|FULL LIST, MORE INFO | |||

4-subnormal subgroup | 2-subnormal subgroup|FULL LIST, MORE INFO | |||

subnormal subgroup | 2-subnormal subgroup|FULL LIST, MORE INFO |

## Facts

- Left residual of 2-subnormal by normal is normal of characteristic: If is a subgroup of with the property that whenever is normal in a group , is 2-subnormal in , then is a normal subgroup of characteristic subgroup in .

## Effect of property operators

### The left transiter

*Applying the left transiter to this property gives*: left-transitively 2-subnormal subgroup