Tarski’s problem for solvable groups

Abstract

In this paper, we show that the free solvable groups (as well as the free nilpotent groups) of finite rank have different elementary theories (i.e., they do not satisfy the same first order sentences of group theory). This result is obtained using a result in group theory (probably due to Malcev and following immediately from a theorem of Auslander and Lyndon) that, for a free nontrivial solvable group, the last nontrivial group in its derived series is is own centralizer.