Universal theories for rigid soluble groups

Abstract

A group is said to be p-rigid, where p is a natural number, if it has a normal series of the form G = G 1 > G 2 > … > G p > G p+1 = 1, whose quotients G i /G i+1 are Abelian and are torsion free when treated as $ \mathbb{Z} $ [G/G i ]-modules. Examples of rigid groups are free soluble groups. We point out a recursive system of universal axioms distinguishing p-rigid groups in the class of p-soluble groups. It is proved that if F is a free p-soluble group, G is an arbitrary p-rigid group, and W is an iterated wreath product of p infinite cyclic groups, then ∀-theories for these groups satisfy the inclusions $ \mathcal{A}(F) \supseteq \mathcal{A}(G) \supseteq \mathcal{A}(W) $ . We construct an ∃-axiom distinguishing among p-rigid groups those that are universally equivalent to W. An arbitrary p-rigid group embeds in a divisible decomposed p-rigid group M = M(α1,…, α p ). The latter group factors into a semidirect product of Abelian groups A 1 A 2…A p , in which case every quotient M i /M i+1 of its rigid series is isomorphic to A i and is a divisible module of rank αi over a ring $ \mathbb{Z} $ [M/M i ]. We specify a recursive system of axioms distinguishing among M-groups those that are Muniversally equivalent to M. As a consequence, it is stated that the universal theory of M with constants in M is decidable. By contrast, the universal theory of W with constants is undecidable.