Abstract
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 n ≥ 2 n \geq 2 the theory of groups solvable of length ≤ n \leq n has no model companion. For the metabelian case ( n = 2 ) (n = 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.
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