Groups, Rings, Logic
Dan Segal (University of Oxford)
Abstract: In group theory, interesting statements about a group usually can’t be ex- pressed 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 profinite 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 $G(R)$.
group theory
Audience: researchers in the topic
GOThIC - Ischia Online Group Theory Conference
Series comments: Please send a message to andrea.caranti@unitn.it to receive a link to the Zoom room where the conference (a series of talks, actually) takes place.
Organizer: | Andrea Caranti* |
*contact for this listing |