**Direct
products**

[6]
Cancellation and elementary equivalence of groups, Journal of Pure
and Applied Algebra 30 (1983), 293-299.

[9]
Produits lexicographiques de groupes ordonnés: Isomorphisme et
équivalence élémentaire, Journal of Algebra 109
(1987), 452-467.

[10]
An example of two nonisomorphic countable ordered abelian groups with
isomorphic lexicographical squares, Bulletin of the London
Mathematical Society 20 (1988), 198-202.

[14]
Cancellation of abelian groups of finite rank modulo elementary
equivalence, Mathematica Scandinavica 67(1990), 5-14.

[15]
Cancellation and elementary equivalence of finitely generated
finite-by-nilpotent groups, Journal of the London Mathematical
Society 44, 1991, 173-183.

[17]
Isomorphism and elementary equivalence of multilinear maps, Linear
and Multilinear Algebra 36 (1994), 151-174.

[18]
The direct decompositions of a group G with G/G' finitely generated,
Transactions of the American Mathematical Society 347, 1995,
1997-2010.

[19]
Model-theoretic properties of polycyclic-by-finite groups, in Model
Theory of Groups and Automorphism Groups, London Mathematical Society
Lecture Note Series 244, Cambridge, 1997.

[20]
Non cancellation and the number of generators, Communications in
Algebra 26 (1998), 35-39.

[21]
Elementary equivalence for finitely generated nilpotent groups and
multilinear maps, Bulletin of the Australian Mathematical Society 58
(1998), 479-493.

If
G is a finite group or a finitely generated abelian group, if
G≈A(1)x...xA(m)≈B(1)x...xB(n), and if the groups A(i),B(j) are
neither trivial nor isomorphic to direct products of non trivial
groups, then we have m=n and there exists a permutation σ of
{1,...,n} such that A(i)≈B(σ(i)) for each iЄ{1,..,n}.

This
property is not true for finitely generated nilpotent groups or
finitely generated finite-by-abelian groups. Even, two decompositions
of the same group do not necessarily have the same number of factors.
Also, two finitely generated finite-by-abelian nilpotent groups A,B
can satisfy AxZ≈BxZ without being isomorphic. R. Hirshon has proved
that two groups G,H which satisfy GxZ≈HxZ have the same finite
images. The converse is true for finitely generated finite-by-abelian
groups according to R.B. Warfield.

More
information concerning this historical background is given in [19],
as well as open questions related to the results discussed below.

In
[6], we show that two groups G,H which satisfy GxZ≈HxZ are
elementarily equivalent. In [14], generalizing [6], we give some
conditions on the abelian groups A,B which imply that any groups G,H
with AxG≈AxH are elementarily equivalent. This property is true, in
particular, if A=B and if S/(S^p) is finite for each subgroup S of A
and each prime number p.

In
[15] and [21], we show that the following properties are equivalent
for two finitely generated finite-by-nilpotent groups G,H:

1)
G and H are elementarily equivalent;

2)
G and H satisfy the same sentences with two alternations of
quantifiers;

3)
GxZ≈HxZ.

The
equivalence of 1) and 3) had been conjectured by V.N. Remeslennikov.
In the case of finitely generated finite-by-nilpotent groups, it
implies a positive answer to the following question, which appears in
page 9 of [L.M. Manevitz, An Abraham Robinson memorial problem list,
Israël Journal of Mathematics 49 (1984), 3-14]: For any groups
G,H and each integer n≥2, is it true that the cartesian product of
n copies of G and the cartesian product of n copies of H are
elementarily equivalent if and only if G and H are elementarily
equivalent?

In
[17] and [21], we obtain similar results for the following classes of
structures, where n≥2 is an integer:

1)
the (n+2)-tuples (A(1),...,A(n+1),f), where A(1),...,A(n+1) are
disjoint finitely generated abelian groups and f:A(1)x...xA(n)→A(n+1)
is an n-linear map;

2)
the triples (A,B,f), where A,B are disjoint finitely generated
abelian groups and f:Ax...xA→B is an n-linear map;

3)
the couples (A,f), where A is a finitely generated abelian group and
f:Ax...xA→A is an n-linear map.

These
results can be applied to integral quadratic forms and Lie rings. In
particular, we show that there exist some nonisomorphic finitely
generated torsion-free Lie rings which are elementarily equivalent.

In
[18], we consider the class C which consists of the groups M with
M/M' finitely generated which satisfy the maximal condition on direct
factors. Any C-group has a decomposition in a finite direct product
of indecomposable groups, and we have seen above that two such
decompositions are not necessarily equivalent up to isomorphism. In
[18], we show that any C-group only has finitely many non equivalent
decompositions. In order to prove this result, we introduce, for
C-groups, a slightly different notion of decomposition, called
J-decomposition; we show that this decomposition is necessarily
unique. As consequences of the properties of J-decompositions, we
obtain several generalizations of results of R. Hirshon. For
instance, we have GxZ≈HxZ for any groups G,H which satisfy GxM≈HxM
for a C-group M.

In
[20], we give an example of two groups G,H, respectively generated by
2 and 3 elements, which satisfy GxZ≈HxZ . The groups G,H are both
extensions of a torsion-free abelian group of rank 3 by an infinite
cyclic group.

In [9] and [10], we consider the lexicographical direct product G*H of two totally ordered abelian groups G,H. We construct some counterexamples which show that implications like (A*A*G≈G)→(A*G≈G) and (G*G≈H*H)→(G≈H) are not true for all A,G,H. On the other hand, these implications become true if we replace elementary equivalence with isomorphism. The results concerning elementary equivalence have been proved partly in [9] and partly by F. Delon and F. Lucas. It is also worth mentioning the results of M. Giraudet on this subject, which make use of [10]. Similar results and counterexamples exist for direct products of abelian groups; the counterexamples have been obtained, notably, by A.L.S. Corner and Eklof-Shelah.