Automorphism group of finitely generated free group

From Groupprops
Jump to: navigation, search


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 \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:

\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.