# I-automorphism

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

## Definition

Suppose $\mathcal{V}$ is a variety of algebras, and $A$ is an algebra in $\mathcal{V}$. An I-automorphism of $A$ is an automorphism that can be expressed as:

$x \mapsto \varphi(x,u_1,u_2,\dots,u_n)$

where $u_1, u_2, \dots, u_n \in A$ are fixed, and $\varphi$ is a word in terms of the operations of the algebra,with the property that for any algebra $B$ of $\mathcal{V}$, and any choice of values $v_1,v_2,\dots,v_n \in B$, the map:

$x \mapsto \varphi(x,v_1,v_2,\dots,v_n)$

gives an automorphism of $B$.

In other words $\varphi$ 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 $x \mapsto gxg^{-1}$. For full proof, refer: Inner automorphisms are I-automorphisms in variety of groups