Wreath Products and Existentially Complete Solvable Groups
Transactions of the American Mathematical Society
It is known that the theory of abelian groups has a model companion but that the theory of groups does not. We show that for any fixed 22 a 2 the theory of groups solvable of length s 22 has no model companion. For the metabelian case (22 = 2) we prove the stronger result that the classes of finitely generic, infinitely generic, and existentially complete metabelian groups are all distinct. We also give some algebraic results on existentially complete metabelian groups. 0. Introduction. In
... troduction. In 1970, Eklof and Sabbagh  showed that the theory 7, of abelian groups has a model companion, but that the theory T of groups does not. Shortly afterward, A. Macintyre  strengthened the negative result for groups by showing that the class of existentially complete groups is distinct from the class of infinitely generic ones. In this paper we consider analogous questions for some theories of groups intermediate between the extremes represented by 7j and 7. Specifically, for each integer n > 1, let T be the theory of groups solvable of length < n. (Some of the model-theoretic and group-theoretic terminology relevant to this paper will be recalled in §1.) Thus 7j gives abelian groups, 7' gives metabelian groups, and so on. Our main results are the following: Theorem 1. For any n>2,T has no model companion. For the case »2 = 2, we have a stronger result: Theorem 2. There is an 3V3 sentence of first-order group theory which holds in every infinitely generic metabelian group and fails in every finitely generic metabelian group. Thus no metabelian group is both finitely generic and infinitely generic, so in particular the class of existentially complete metabelian groups is distinct from the class of infinitely generic ones and from the class of finitely generic ones.