Our first goal is to study the elementary theory of certain group extensions and apply these results to study the classical linear groups PSL(2;Z), SL(2; Z), PGL(2;Z) and GL(2; Z) and describe finitely generated groups elementarily equivalent to any of the aforementioned groups. The importance of studying the first-order properties of these groups was stressed in the seminal work of A. Malcev on the elementary theory of linear groups.

The problem of understanding the elementary theory of a free pro-p group is currently among the most well-known and important open problems in model theory of groups. Our second goal is to begin a systematic study of the first-order theory of a free pro-p group of finite rank.