Model-theoretic properties of polycyclic-by-finite groups

Summary

If R is a (possibly noncommutative) ring, then, by [P, Corollary 2.18, p. 37], any R–module M is characterized up to elementary equivalence by the invariants, where and ∣φ(M)/(φ(M) ⋂ ψ(M))∣ ∈ {1, 2, …, ∞}, are positive primitive formulas with one free variable. Some algebraic invariants which characterize abelian groups up to elementary equivalence, and which can be written in the form ∣φ(M)/(φ(M) ⋂ ψ(M))∣ had been previously given by W. Szmielew and by P.C. Eklof and E.R. Fisher (see [EF]). The two following consequences are easily proved:

1. 1) Two abelian groups, or two modules, M,N, are elementarily equivalent if and only if they satisfy the same sentences with one alternation of quantifiers.

2. 2) For each integer n ≥ 2, two abelian groups, or two modules, M, N, are elementarily equivalent if and only if the direct product of n copies of M and the direct product of n copies of N are elementarily equivalent.

For nonabelian groups in general, it is not possible to obtain such a characterization of elementary equivalence, since S. Burris proved in [Bu] that, for each integer n, there exist two soluble groups which satisfy the same sentences with n alternations of quantifiers without being elementarily equivalent. Concerning 2), L. Manevitz proposes the following problem in [Mn, p. 9]:

Conjecture.For each integer n ≥ 2, two groups M,N are elementarily equivalent if and only if the direct product of n copies of M and the direct product of n copies of N are elementarily equivalent.