a b s t r a c tIn this paper, we give a complete algebraic description of groups elementarily equivalent to the P. Hall completion of a given free nilpotent group of finite rank over an arbitrary binomial domain. In particular, we characterize all groups elementarily equivalent to a free nilpotent group of finite rank.

Keywords: 2-nilpotent group of finite rank, elementarily equivalent groups.We give a characterization of groups elementarily equivalent to a free 2-nilpotent group of finite rank. PRELIMINARIESIn this paper we give an algebraic description of groups elementarily equivalent to a given free nilpotent class 2 group of an arbitrary finite rank. An elementary classification problem for groups.A. Tarski and A. I. Mal'tsev's fundamental work in model theory enlightened on importance of an elementary (logical) classification of algebraic structures. The general elementary classification problem insists on describing (in algebraic terms) all algebraic structures (perhaps, from a given class) that are elementarily equivalent to a given one. Recall that two algebraic structures A and B in a language L are elementarily equivalent (A ≡ B) if the two structures have the same first-order theories in L (indistinguishable in the first-order logic in L).Tarski in [1] was concerned with elementary theories for an algebraically closed field R and for a real closed field C. Subsequently, J. Ax and S. Kochen [2-4] and Yu. L. Ershov [5-7] studied into elementary theories for fields Q p of p-adic numbers, which led to the creation of a general theory of algebraically closed, real closed, and p-adically closed fields. These results became classical in algebra and can be found in many monographs in algebra and model theory.A first important result on elementary classification of groups is due to W. Szmielew-she classified elementary theories of Abelian groups in terms of so-called Szmielew invariants [8] (see also [9][10][11]). For non-Abelian groups, the main inspiration, probably, was Tarski's famous problem inquiring whether or not free non-Abelian groups of finite ranks are elementarily equivalent. This problem had been open for many years and only recently was solved positively in [12,13]. In fact, free non-Abelian groups of finite rank turned out to be elementarily equivalent. The situation is different for free solvable (or nilpotent) groups of finite rank. Such groups are elementarily equivalent iff they are isomorphic [14]. Indeed, in these cases the Abelianization G/ [G, G] (and, hence, the rank) of a group G is definable (interpretable) in G by first-order formulas, whence the result.In [15] Mal'tsev described elementary equivalence for classical linear groups. Namely, it was shown that if G ∈ {GL, P GL, SL, P SL}, n, m 3, where K and F are fields of characteristic zero, then G(F )

Abstract. In this paper we further study properties of Magnus embedding, give a precise reducibility diagram for Dehn problems in groups of the form F/N (d) , and provide a detailed answer to Problem 12.98 in Kourovka notebook. We also show that most of the reductions are polynomial time reductions and can be used in practical computation.

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In this paper we initiate a study of first-order rich groups, i.e., groups where the first-order logic has the same power as the weak second order logic. Surprisingly, there are quite a lot of finitely generated rich groups, they are somewhere in between hyperbolic and nilpotent groups (these ones are not rich). We provide some methods to prove that groups (and other structures) are rich and describe some of their properties. As corollaries we look at Malcev's problems in various groups.