# Inner automorphism to automorphism is right tight for normality

## Contents

## Statement

### Property-theoretic statement

Consider the subgroup property of being a Normal subgroup (?). Then, the following function restriction expression for this subgroup property:

Inner automorphism Automorphism

is a right tight function restriction expression.

### Statement with symbols

Suppose is a group and is a function. Consider the following property for :

There exists a group containing as a normal subgroup, and an inner automorphism of such that the restriction of of .

This property is *precisely* equivalent to being an automorphism. (Note that by definition of normality, any such must be an automorphism, so the content of the theorem is that every automorphism can be realized this way).

## Facts used

- Every group is normal fully normalized in its holomorph: If is a group, there is a group (the holomorph of ) containing as a normal subgroup, with the property that
*every*automorphism of extends to an inner automorphism in .

## Related facts

### Other facts using this idea or similar proofs

- Left transiter of normal is characteristic: If is such that whenever is normal in , is also normal in , then is characteristic in . This fact follows from the statement on this page and the right tightness theorem.
- Every group is normal fully normalized in its holomorph
- Normal upper-hook fully normalized implies characteristic

If we drop the requirement that be normal in , then the result is no longer true. In fact, we have:

- Every injective endomorphism arises as the restriction of an inner automorphism
- Isomorphic iff potentially conjugate: Two subgroups of a group that are isomorphic are conjugate in some bigger group containing it.
- Isomorphic iff potentially conjugate in finite: Two subgroups of a finite group that are isomorphic are conjugate in some bigger finite group containing it.

## Proof

### Proof using the holomorph

The proof using the holomorph finds a *single* big group that works for all automorphisms.

**Given**: A group , an automorphism of .

**To prove**: There exists a group containing as a normal subgroup, such that extends to an inner automorphism of .

**Proof**: Let be the holomorph of . Fact (1) now completes the proof.

### Proof using semidirect product with just that automorphism

This proof uses a more minimalistic construction, but the construction differs for every automorphism. This proof technique generalizes to proving that every injective endomorphism arises as the restriction of an inner automorphism.

**Given**: A group , an automorphism of .

**To prove**: There exists a group containing as a normal subgroup, such that extends to an inner automorphism of .

**Proof**: Consider the group , where the generator of acts on by . is normal in the semidirect product by definition, and the automorphism extends to the inner automorphism of induced by conjugation by that generator of .