Finite images and elementary equivalence

[2] Images finies et équivalence élémentaire de groupes et autres structures algébriques, Thèse de doctorat d'Etat, Université Paris VII, 20 October 1986.
[4] Des groupes nilpotents de classe 2 sans torsion de type fini ayant les mêmes images finies peuvent ne pas être élémentairement équivalents, Comptes Rendus de l'Académie des Sciences de Paris 294 (1982), 1-4.
[5] Equivalence élémentaire entre groupes finis-par-abéliens de type fini, Commentarii Mathematici Helvetici 57 (1982), 469-480.
[7] The model theory of finitely generated finite-by-abelian groups, Journal of Symbolic Logic 49 (1984), 1115-1124.
[8] Elementary equivalence and isomorphism of finitely generated nilpotent groups, Communications in Algebra 12 (1984), 1899-1915.
[11] Elementary equivalence and genus of finitely generated nilpotent groups, Bulletin of the Australian Mathematical Society 37 (1988), 61-68.
[12] Elementary equivalence and profinite completions: A characterization of finitely generated abelian-by-finite groups, Proceedings of the American Mathematical Society 103 (1988), 1041-1048.
[13] Elementary equivalence of a polycyclic-by-finite group and its profinite completion, Archiv der Mathematik 52 (1989), 521-525.
[16] Finite images and elementary equivalence of completely regular inverse semigroups and other diagrams of groups, Semigroup Forum 45, 1992, 322-331.

The finite images of a group G are the finite groups H such that there exists a surjective homomorphism from G to H. The profinite topology of G is the uniform topology defined by taking the set of all normal subgroups of finite index as a fundamental system of neighbourhoods of 1. The completion of G for this topology is called profinite completion.
For each group G and each integer n≥1, we denote by G^n the subgroup generated by the n-th powers of elements of G. If G is polycyclic-by-finite, then each G/(G^n) is finite; moreover, G is naturally embedded in its profinite completion, because the intersection of its normal subgroups of finite index is {1}.

The following problem was proposed in 1964 by K.A. Hirsch: Is a polycyclic-by-finite group completely determined by its finite images?
The answer is positive for finitely generated abelian groups. On the other hand, there are various examples of finitely generated nilpotent groups or finitely generated finite-by-abelian groups which have the same finite images without being isomorphic. Anyhow, F.J. Grünewald, P.F. Pickel and D. Segal have proved that any class of polycyclic-by-finite groups which have the same finite images is a finite union of isomorphism classes.
The first section of [2] gives more information concerning this historical background.

The definability of the subgroups M^n in a polycyclic-by-finite group M is an essential argument in the proofs of the results mentioned below.

In [5], we show that, for each polycyclic-by-finite group G and each elementary extension S of G which satisfies the countable 1-types with parameters in G, there exists a natural isomorphism from S/E(S) to the profinite completion of G, where E(S) is the intersection of the subgroups S^n. It follows that two elementarily equivalent polycyclic-by-finite groups have isomorphic profinite completions, and therefore have the same finite images. We also prove that the converse is true for finitely generated finite-by-abelian groups (this result is generalized in [11]).
In [7], we use the results of [5] and the properties of profinite completions in order to characterize the models of the theory of a finitely generated finite-by-abelian group, the elementary embeddings between these models, and the saturated models.
[4] and [8] give families of examples of finitely generated torsion-free nilpotent groups which have the same finite images without being elementarily equivalent. The examples in [4] are interesting because the groups satisfy stronger similarity properties (they have the same ''genus'').

In [11], we show that, if two finitely generated finite-by-nilpotent groups G,H satisfy the same sentences with one alternation of quantifiers, then each of them is isomorphic to a subgroup of finite index of the other one, and this index may be chosen prime to any given integer. In particular, G and H have the same finite images, the same Pickel genus and the same Mislin genus. Later on, a generalization of this result was obtained by D. Raphael.
In [12], we prove that any finitely generated abelian-by-finite group is an elementary submodel of its profinite completion. It follows that two finitely generated abelian-by-finite groups are elementarily equivalent if and only if they have the same finite images.
On the other hand, we show in [13] that, for each polycyclic-by-finite group G which is not abelian-by-finite, there is an existential sentence which is false in G and true in the profinite completion of G.
In [16], we generalize the methods of [12] in order to prove that two diagrams of finitely generated abelian-by-finite groups are elementarily equivalent if and only if they have the same finite images (here, a diagram of groups is a structure which consists of one or several groups and one or several homomorphisms between these groups). In particular, we show that there exist:
- two endomorphisms f,g of ZxZ such that (ZxZ,f) and (ZxZ,g) are elementarily equivalent, but not isomorphic;
- four subgroups S(1),S(2),T(1),T(2) of ZxZ such that ( ZxZ,S(1),S(2)) and (ZxZ,T(1),T(2)) are elementarily equivalent, but not isomorphic;
- two nonisomorphic finitely generated torsion-free commutative semigroups which are elementarily equivalent.

In section IX of [2], we show that several results proved in [2] or previously known, which use symmetrical properties like GxZ≈HxZ or ''G and H have the same finite images'' or ''G and H are elementarily equivalent'', can be generalized by considering non symmetrical properties like: ''there exists a surjective homomorphism from GxZ to HxZ'' or ''any finite group which is an image of H is also an image of G'' or ''any positive sentence which is true in G is also true in H''.
The other results in [1], [2], [3] are not analyzed here since they also appear in other papers.