**Axiomatization
and other model-theoretic properties of groups**

[22]
An application of Ramsey's theorem to groups with homogeneous theory,
Communications in Algebra 28 (2000), 2977-2981.

[23]
Axiomatization of abelian-by-G groups for a finite group G, Archive
for Mathematical Logic 40 (2001), 515-521.

[24]
Elementary equivalence for abelian-by-finite and nilpotent groups,
Journal of Symbolic Logic 66 (2001), 1471-1480.

[26]
Quasi-finitely axiomatizable nilpotent groups (with G. Sabbagh),
Journal of Group Theory 9 (2006), 95-106.

[27]
Quasi-finitely axiomatizable groups and groups which are prime
models, Journal of Group Theory 9 (2006), 107-116.

[30]
Some new examples of quasi-finitely axiomatizable groups which are
prime models, 5 pages, in preparation.

In
[22], for any integers n>m≥2, we say that a complete theory T is
**(m,n)-homogeneous** if,
for each model M of T, any n-tuples a,b in M have the same type if
the corresponding m-tuples from a,b have the same type. It was
conjectured by H. Kikyo that, if M is an infinite group, with
possibly additional structure, then the theory of M is not
(m,n)-homogeneous. We prove a general result on structures with
(m,n)-homogeneous theory which implies that, if M is a
counterexample to this conjecture, then there exists an integer h
such that each abelian subgroup of M has at most h elements. It
follows that there exist an integer k such that M^k=1, and an
integer l such that each finite subgroup of M has at most l elements.

In
[23], we show that, for each finite group G, there exists an
axiomatization of the class of abelian-by-G groups with a unique
sentence. This result had been conjectured by A. Marcja and C.
Toffalori. In the proof, we use the definability of the subgroups M^n
in an abelian-by-finite group M, and the Auslander-Reiten sequences
for modules over an Artin algebra.

In
[24], we show that two abelian-by-finite groups are elementarily
equivalent if and only if they satisfy the same sentences with two
alternations of quantifiers. We also prove that abelian-by-finite
groups satisfy
a quantifier elimination property. On the other hand, for each
integer n, we give some examples of nilpotent groups which satisfy
the same sentences with n alternations of quantifiers and do not
satisfy the same
sentences with n+1 alternations of quantifiers.

More
recently, new techniques introduced by A. Morozov et A. Nies made it
possible to have a better insight into the model-theoretic properties
of finitely generated groups. According to their definition, we say
that a finitely generated group G is **quasi-finitely
axiomatizable**, or more briefly
**QFA**, if it satisfies
a sentence φ such that each finitely generated group H which
satisfies φ is isomorphic to G. We say that G is a **prime
model** of its theory, or more
briefly that G is **prime**,
if it is isomorphic to an elementary submodel of any model of its
theory.

In
[26] and [27], we investigate the relations between these two
properties, and their algebraic consequences. For finitely generated
nilpotent-by-finite groups, we prove that the two properties are
equivalent and we characterize them algebraically. It follows that
finitely generated free nilpotent groups and groups of upper
unitriangular nxn matrices for n≥3 are QFA, which gives a positive
answer to a question of A. Nies, and that they are prime, which
disproves a conjecture of O.V. Belegradek.

In
[30], we consider groups which are semi-direct products of a finitely
generated torsion-free abelian group by a cyclic group. We
investigate the following questions: Are they quasi-finitely
axiomatizable (QFA)? Are they prime models of their theory? Are they
elementarily equivalent?

It
follows from the results of A. Morozov et A. Nies that there exist
some finitely generated prime groups which are not QFA. On the other
hand, we do not know presently if a QFA group is necessarily prime.
Also one can wonder whether there exists an infinite QFA group, or
more generally an infinite finitely generated group, with a finitely
axiomatizable theory.