Abstract

A group is n-free if every subgroup generated by n or fewer distinct elements is free. In [8], the authors observed that if G is a finitely generated model of the elementary theory of the non-Abelian free groups, then G is 2-free. The main result here is that such a group is 3-free. The principal tool used is a characterization, due to Hmelevskii [10], of the set of nontrivial solutions in a free group of a three variable word equation without coefficients.

Introduction and preliminaries

We start by giving a convention and definition which are used throughout this paper.

Convention. The trivial group {1} is free of rank zero.

Definition 1. Let n be a positive integer. The group G is n-free provided every subgroup of G generated by n or fewer distinct elements is free.

Clearly, every n-free group is m-free for all integers m with 1 ≤ m ≤ n. The 1-free groups are precisely the torsion free groups.

Lemma 1. (Harrison [9]) Let G be a group. Then the following three properties are pairwise equivalent.

1. (i) The relation of commutativity is transitive on the non-identity elements of G.

2. (ii) The centralizer in G, ZG(g), of every non-identity element g ≠ 1 in G is Abelian.

3. (iii) Every pair of distinct maximal Abelian subgroups M1 ≠ M2 in G has a trivial intersection; M1 ∩ M2 = {1}.