La classification élémentaire de groupes abeliens (Q789519)

The author characterizes the classes of abelian groups which are elementarily equivalent in the sense of logic. The answer is found on p. 28: If \(\phi\) (x) is a formula of the theory in one free variable x such that \(\phi\) (x) and \(\phi\) (y) imply \(\phi\) (x-y) then \(H=\{a\in A:\quad \phi(a)\quad holds\quad in\quad A\}\) is a subgroup of A for any abelian group A. If K is another subgroup determined by some formula then the cardinalities of the quotients H/\(H\cap K\) provide the crucial invariants. Two abelian groups A and B are elementarily equivalent if and only if each quotient H/\(H\cap K\) in A has the same finite cardinality as the corresponding quotient in B or both are infinite. It is actually not necessary to consider all such quotients H/\(H\cap K\) but only those with H and K in the subgroup lattice \({\mathcal L}(A)\) which is generated by A and \(\{\) 0\(\}\) under the operations \(\cap\), \(+\), and those of taking images and preimages under the endomorphisms \(x\mapsto nx\) where n is a natural number. This description is handy, e.g. for any non-zero divisible torsion-free group A we have \({\mathcal L}(A)=\{A,0\}\) and hence any two such groups are elementarily equivalent. In addition to the above result it is shown that the first order theory of abelian groups is decidable. The paper is well-written and supplies informally the logical concepts required.