Automorphism group of finitely generated free group

Definition

Let $n$ be a natural number. The automorphism group of the free group on $n$ generators is defined as the automorphism group of $F_n$, the free group on $n$ generators. A group is termed the automorphism group of a finitely generated free group if it is isomorphic to $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$ for some natural number $n$.

Relation with other groups

General relation with reduced free groups

Let $\mathcal{V}$ be any subvariety of the variety of groups. If $F_n(\mathcal{V})$ denotes the free algebra on $n$ generators in $\mathcal{V}$, then $F_n(\mathcal{V})$ is the quotient of $F_n$ by a verbal subgroup. Groups of the form $F_n(\mathcal{V})$ are termed reduced free groups.

There is a natural homomorphism from $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$ to $\mathrm{A}\mathrm{u}\mathrm{t}(F_n(\mathcal{V}))$, which sends an automorphism of the free group to the induced automorphism of $F_n(\mathcal{V})$. One way of seeing this is to observe that $F_n(\mathcal{V})$ is the quotient of $F_n$ by a verbal subgroup, which is in particular a characteristic subgroup, hence any automorphism of $F_n$ descends to an automorphism of the quotient.

However, this homomorphism is not necessarily surjective. For instance, if $\mathcal{V}$ is given as the variety of abelian groups where every element has order $p$ for some prime $p\ge 5$, there are automorphisms of $F_1(\mathcal{V}=\mathbb{Z}/p\mathbb{Z}$ that do not arise from automorphisms of $F_1=\mathbb{Z}$.

Relation with free abelian groups

Free abelian groups are free algebras in the variety of abelian groups. The free abelian group of rank $n$ is isomorphic to ${\mathbb{Z}}^n$, and its automorphism group is isomorphic to $GL(n,\mathbb{Z})$. Thus, we have a homomorphism:

$\mathrm{A}\mathrm{u}\mathrm{t}(F_n)\to GL(n,\mathbb{Z})$.

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

IAPS structure

Further information: IAPS of automorphism groups of free groups

The collection of groups $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$ form an IAPS of groups. In other words, we can construct injective homomorphisms:

${\mathrm{\Phi }}_{m,n}:\mathrm{A}\mathrm{u}\mathrm{t}(F_m)\times \mathrm{A}\mathrm{u}\mathrm{t}(F_n)\to \mathrm{A}\mathrm{u}\mathrm{t}(F_{m+n})$

satisfying the associativity condition.