For a given family of groups B, we define an elementary class E(B) as the smallest class of groups, that contains B and is closed under the passing to subgroups, quotients, and taking extensions and direct limits. The properties of these classes are considered. The proposed theory has close connections with the theory of elementary amenable groups as well as of Kuroš – Černikov classes.

We compute the cobordism rings of the classifying spaces of a certain class of Chevalley groups. In the particular case of the general linear group, we prove that it is isomorphic to the Chow ring.

A fundamental theorem of Barsotti and Chevalley states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth affine algebraic group. In 1956 Rosenlicht gave a short proof of the theorem. We explain his proof in the language of modern algebraic geometry.