# Nonstandard definitions of inner automorphism

This article lists nonstandard definitions of the following term: inner automorphism

This page lists some offbeat, but correct, definitions of inner automorphism.

## Contents

## In terms of being able to extend to bigger groups

### Extensible automorphism

`Further information: Extensible equals inner, Inner implies extensible`

An automorphism of a group is termed **inner** if it is an extensible automorphism: For any group containing , extends to an automorphism of .

### Pushforwardable automorphism

`Further information: Pushforwardable equals inner`

An automorphism of a group is termed **inner** if it is a pushforwardable automorphism: For any homomorphism of groups , there is an automorphism of such that .

### Quotient-pullbackable automorphism

`Further information: Quotient-pullbackable equals inner, Inner implies quotient-pullbackable`

An automorphism of a group is termed **inner** if it is a quotient-pullbackable automorphism: For any surjective homomorphism , there exists an automorphism of such that .

## Definition in terms of universal algebra

### I-automorphism

`Further information: Inner automorphisms are I-automorphisms in the variety of groups`

An automorphism of a group is termed an inner automorphism if it is an I-automorphism in the variety of groups.