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.
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Supported by RFBR, project No. 09-01-00099.
Translated from Algebra i Logika, Vol. 50, No. 6, pp. 802-821, November-December, 2011.
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Myasnikov, A.G., Romanovskii, N.S. Universal theories for rigid soluble groups. Algebra Logic 50, 539–552 (2012). https://doi.org/10.1007/s10469-012-9164-y
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DOI: https://doi.org/10.1007/s10469-012-9164-y