On some model theoretic problems concerning certain extensions of abelian groups by groups of finite exponent (Q1589928)

This paper is devoted to the study of the following classes of groups derived from a fixed group \(G\). Let \(\mathfrak{K} (G)\) be the class of all groups \(S\) with abelian normal subgroup \(A\) such that \(S/A\) and \(G\) are elementary equivalent \((G \equiv S/A)\). Moreover, let \(\mathfrak{K}_{a b} (G)= \{X \in \mathfrak{K} (G)\), \(X\) abelian\}. \textit{F. Oger} [``Axiomatization of the class of abelian-by-\(G\)-groups for a finite group \(G\)'' (to appear)] showed hat \(\mathfrak{K} (G)\) is finitely axiomatizable for finite \(G\). This motivated the author to consider infinite groups and to see when \(\mathfrak{K} (G)\) is elementary or finitely axiomatizable. Special attention is given to the case when \(G\) is the direct sum of a finite group and an elementary abelian \(p\)-group of rank \(\aleph_0\). In this particular case \(\mathfrak{K}_{ab} (G)\) can be characterized and cases can be singled out when \(\mathfrak{K} (G)\) is elementary.