# Pushforwardable equals inner

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term inner automorphism

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

This fact is related to: Extensible automorphisms problem

View other facts related to Extensible automorphisms problemView terms related to Extensible automorphisms problem |

## Contents

## Statement

The following are equivalent for an automorphism of a group :

- is a pushforwardable automorphism of : For any homomorphism , there exists an automorphism of such that .
- is an inner automorphism.

## Definitions used

### Pushforwardable automorphism

An automorphism of a group is termed a pushforwardable automorphism if, for any homomorphism , there exists an automorphism of such that .

### Inner automorphism

`Further information: Inner automorphism`

## Facts used

- Extensible equals inner: An automorphism of a group is inner if and only if it can be extended to an automorphism for any group containing .

## Proof

### Proof that inner implies pushforwardable

If for , and is a homomorphism, we can take .

### Proof that pushforwardable implies inner

If is pushforwardable, then, in particular, is extensible in the sense that it can be extended to an automorphism for any group containing . Thus, by fact (1), is inner.