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 $\operatorname {Aut} (F_{n})$ 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 $\operatorname {Aut} (F_{n})$ to $\operatorname {Aut} (F_{n}({\mathcal {V}}))$, 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 $p\geq 5$, 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:

$\operatorname {Aut} (F_{n})\to GL(n,\mathbb {Z} )$.

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 $\operatorname {Aut} (F_{n})$ form an IAPS of groups. In other words, we can construct injective homomorphisms:

$\Phi _{m,n}:\operatorname {Aut} (F_{m})\times \operatorname {Aut} (F_{n})\to \operatorname {Aut} (F_{m+n})$

satisfying the associativity condition.