# Every subgroup is contracharacteristic in its normal closure

From Groupprops

This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Contracharacteristic subgroup (?) and Normal subgroup (?)), to another known subgroup property (i.e., Subgroup (?))

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This fact is an application of the following pivotal fact/result/idea:characteristic of normal implies normal

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## Contents

## Statement

Any subgroup of a group is a contracharacteristic subgroup of its normal closure. In particular, it occurs as a contracharacteristic subgroup of a normal subgroup.

## Definitions used

### Contracharacteristic subgroup

`Further information: Contracharacteristic subgroup`

A subgroup of a group is termed **contracharacteristic** if it is not contained in any proper characteristic subgroup.

## Related facts

- Characteristic of normal implies normal
- Equivalence of definitions of subgroup of Abelian normal subgroup

## Facts used

## Proof

### Hands-on proof

**Given**: A subgroup , is the normal closure of in .

**To prove**: If is a characteristic subgroup of containing , then .

**Proof**:

- By fact (1), we see that since is characteristic in and is normal in , we obtain that is normal in .
- Thus, is a normal subgroup of containing . By definition of normal closure, we get that . Since , we get .