# Inner automorphisms are I-automorphisms in the variety of groups

## Contents

## Statement

In the Variety of groups (?), treated as a variety of algebras, an I-automorphism (?) is the same thing as an Inner automorphism (?).

## Definitions used

### Inner automorphism

`Further information: Inner automorphism`

An automorphism of a group is termed an inner automorphism if there exists such that , where we define:

### I-automorphism

`Further information: I-automorphism`

Suppose is a variety of algebras, and is an algebra in . An **I-automorphism** of is an automorphism that can be expressed as:

where are fixed, and is a word in terms of the operations of the algebra,with the property that for *any* algebra of , and *any* choice of values , the map:

gives an automorphism of .

In other words is *guaranteed* to give an automorphism.

## Facts used

- Inner automorphisms actually
*are*automorphisms.`For full proof, refer: Group acts as automorphisms by conjugation`

## Proof

### Inner automorphisms are I-automorphisms

can be viewed as a word with input and parameter . This word gives an automorphism for *every* group and every choice of parameter . Thus, inner automorphisms are I-automorphisms.

### I-automorphisms are inner automorphisms

**Given**: A word with the property that for any group and any choice of values of in , the map sending to is an automorphism

**To prove**: This automorphism is always inner.

**Proof**: Let be the free group on generators, and let be a freely generating set for . By the given condition, the map:

gives an automorphism of . In particular:

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A little manipulation of possible expressions shows that must be of the form:

Hence, any automorphism obtained using must be an inner automorphism.