An elementary class extending abelian-by-\(G\) groups, for \(G\) infinite (Q1357129)

For any finite group \(G\), the class of abelian-by-\(G\) groups is elementary; this was proved by A. Marcja and the author for any \(G\) of square-free order and by F. Oger in the general case. In the paper under review it is shown that this fails for every infinite group \(G\). Let \(\mathcal{K}(G)\) be the class of all groups elementarily equivalent to \(G\). The author shows that, for any abelian group \(G\) of prime exponent, the class of all abelian-by-\(\mathcal{K}(G)\) groups is elementary. On the other hand, he gives a series of examples of groups \(G\), for which the result does not hold.