# I-automorphism

From Groupprops

This article defines a property that can be evaluated for an automorphism of an algebra in a variety of algebras. The evaluation of that property depends on the ambient variety, and not just on the automorphism or the algebra.

View all such properties

## Contents

## Definition

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.

## Particular cases

### For groups

In the variety of groups, the I-automorphisms are precisely the inner automorphisms: the automorphisms of the form . `For full proof, refer: Inner automorphisms are I-automorphisms in variety of groups`