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), 14.
[5]
Equivalence élémentaire entre groupes
finisparabéliens de type fini, Commentarii Mathematici
Helvetici 57 (1982), 469480.
[7]
The model theory of finitely generated finitebyabelian groups,
Journal of Symbolic Logic 49 (1984), 11151124.
[8]
Elementary equivalence and isomorphism of finitely generated
nilpotent groups, Communications in Algebra 12 (1984), 18991915.
[11]
Elementary equivalence and genus of finitely generated nilpotent
groups, Bulletin of the Australian Mathematical Society 37 (1988),
6168.
[12]
Elementary equivalence and profinite completions: A characterization
of finitely generated abelianbyfinite groups, Proceedings of the
American Mathematical Society 103 (1988), 10411048.
[13]
Elementary equivalence of a polycyclicbyfinite group and its
profinite completion, Archiv der Mathematik 52 (1989), 521525.
[16]
Finite images and elementary equivalence of completely regular
inverse semigroups and other diagrams of groups, Semigroup Forum 45,
1992, 322331.
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 nth powers of elements of G. If G is
polycyclicbyfinite, 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
polycyclicbyfinite 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 finitebyabelian 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 polycyclicbyfinite 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 polycyclicbyfinite group M
is an essential argument in the proofs of the results mentioned
below.
In
[5], we show that, for each polycyclicbyfinite group G and each
elementary extension S of G which satisfies the countable 1types
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
polycyclicbyfinite groups have isomorphic profinite completions,
and therefore have the same finite images. We also prove that the
converse is true for finitely generated finitebyabelian 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 finitebyabelian group, the elementary embeddings
between these models, and the saturated models.
[4]
and [8] give families of examples of finitely generated torsionfree
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 finitebynilpotent
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 abelianbyfinite group is
an elementary submodel of its profinite completion. It follows that
two finitely generated abelianbyfinite 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 polycyclicbyfinite
group G which is not abelianbyfinite, 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 abelianbyfinite 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 torsionfree 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.