Automorphism group of finitely generated free group
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
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.
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.