Семинары ИМ СО РАН

In group theory, interesting statements about a group usually can't be expressed in the language of first-order logic. It turns out, however, that some groups can actually be determined by their first-order properties, or, even more strongly, by a single first-order sentence. In the latter case the group is said to be finitely axiomatizable.

I will describe some examples of this phenomenon (joint work with A. Nies and K. Tent). One family of results concerns axiomatizability of $p$-adic analytic pro-$p$ groups, within the class of all rofinite groups. Another main result is that for an adjoint simple Chevalley group of rank at least 2 and an integral domain $R$; the group $G(R)$ is bi-interpretable with the ring $R$. This means in particular that first-order properties of the group $G(R)$ correspond to first-order properties of the ring $R$. As many rings are known to be finitely axiomatizable we obtain the corresponding result for many groups; this holds in particular for every finitely generated group of the form $G(R)$.