Automorphism groups of relatively free groups

Abstract

For each positive integer n, let Fn be a free group of rank n with basis (in other words, free generating set) {x1, …, xn} . If θ is an automorphism of Fn then {x1θ, …, xnθ} is also a basis of Fn and every basis of Fn has this form.For any variety of groups [Vfr ], let [Vfr ](Fn) denote the verbal subgroup of Fn corresponding to [Vfr ]. (See [10] for information on varieties and related concepts.) Let Gn = Fn/[Vfr ](Fn). Then Gn is a relatively free group of rank n in the variety [Vfr ]. By a basis of Gn we mean a subset S such that every map of S into Gn extends, uniquely, to an endomorphism of Gn. Write x¯i = xi[Vfr ](Fn) for i = 1, …, n. Then {x¯1, …, x¯n} is a basis of Gn. If λ is an automorphism of Gn then {x¯1λ, …, x¯nλ} is also a basis of Gn and every basis of Gn has this form.Any automorphism θ of Fn induces an automorphism θ of Gn in which x¯iθ = (xiθ)[Vfr ](Fn) for i = 1, …, n. Thus every basis of Fn induces a basis of Gn. The converse however is not always true; in general, there are automorphisms of Gn which are not induced by automorphisms of Fn.