# Equivalence of definitions of transitively normal subgroup

This article gives a proof/explanation of the equivalence of multiple definitions for the term transitively normal subgroup

View a complete list of pages giving proofs of equivalence of definitions

## Contents

## Statement

The following are equivalent for a normal subgroup of a group :

- For any normal subgroup of , is a normal subgroup of .
- For any normal automorphism of , the restriction of to is also a normal automorphism of .

## Definitions used

### Normal automorphism

`Further information: Normal automorphism (?)`

An automorphism of a group is termed a **normal automorphism** if, for every normal subgroup of , restricts to an automorphism of .

## Proof

### (1) implies (2)

**Given**: A group , a subgroup of such that any normal subgroup of is also a normal subgroup of . is a normal automorphism of .

**To prove**: restricts to an automorphism of that is a normal automorphism of .

**Proof**:

- is normal in : Since is a normal subgroup of itself, setting in the condition on yields that is normal in .
- restricts to an automorphism of : This follows from the previous step and the definition of normal automorphism.
- For any normal subgroup of , is a normal subgroup of : This is by assumption.
- For any normal subgroup of , restricts to an automorphism of : By the previous step, is normal in , so, by assumption, restricts to an automorphism of . But since and the restriction of to is , the restriction of to equals the restriction of to .

From the last step, is a normal automorphism of , completing the proof.

### (2) implies (1)

**Given**: A group , a subgroup of such that every normal automorphism of restricts to a normal automorphism of . is a normal subgroup of .

**To prove**: is a normal subgroup of , i.e., for any and , .

**Proof**: Let be conjugation by (i.e., an inner automorphism).

- is a normal automorphism of : By the definition of normal subgroup, restricts to an automorphism of every normal subgroup of . Thus, is a normal automorphism of .
- The restriction of to , which we call , is a normal automorphism of : This follows from the given data for .
- and hence, , restricts to an automorphism of : Since is a normal subgroup of , and is a normal automorphism of , restricts to an automorphism of . Hence, restricts to an automorphism of .
- , i.e., : This follows immediately from the previous step.