# Finite axiomatizability for profinite groups

## TL;DR

This paper investigates which profinite groups can be uniquely characterized by a single first-order sentence within the class of profinite groups, establishing finite axiomatizability results and discussing limitations.

## Contribution

It proves that various classes of profinite groups are finitely axiomatizable in the class of profinite groups, advancing understanding of their logical definability.

## Key findings

- Certain profinite groups are finitely axiomatizable within their class.
- Some groups cannot be finitely axiomatized, with reasons discussed.
- The paper provides criteria for finite axiomatizability in profinite groups.

## Abstract

A group is $\textit{finitely axiomatizable}$ (FA) in a class $\mathcal{C}$ if it can be determined up to isomorphism within $\mathcal{C}$ by a sentence in the first-order language of group theory. We show that profinite groups of various kinds are FA in the class of profinite groups. Reasons why certain groups cannot be FA are also discussed.