# Automorphism group of finitely generated free group

## Contents

## Definition

Let be a natural number. The **automorphism group of the free group** on generators is defined as the automorphism group of , the free group on generators. A group is termed the **automorphism group of a finitely generated free group** if it is isomorphic to for some natural number .

## Relation with other groups

### General relation with reduced free groups

Let be any subvariety of the variety of groups. If denotes the free algebra on generators in , then is the quotient of by a verbal subgroup. Groups of the form are termed reduced free groups.

There is a natural homomorphism from to , which sends an automorphism of the free group to the induced automorphism of . One way of seeing this is to observe that is the quotient of by a verbal subgroup, which is in particular a characteristic subgroup, hence any automorphism of descends to an automorphism of the quotient.

However, this homomorphism is not necessarily surjective. For instance, if is given as the variety of abelian groups where every element has order for some prime , there are automorphisms of that do not arise from automorphisms of .

### Relation with free abelian groups

Free abelian groups are free algebras in the variety of abelian groups. The free abelian group of rank is isomorphic to , and its automorphism group is isomorphic to . Thus, we have a homomorphism:

.

It turns out that this automorphism is surjective -- in other words, every automorphism of the free abelian group on generators arises from an automorphism of the free group on generators.

## IAPS structure

`Further information: IAPS of automorphism groups of free groups`

The collection of groups form an IAPS of groups. In other words, we can construct injective homomorphisms:

satisfying the associativity condition.