Finite axiomatizability for profinite groups

Abstract

A group is $\text{finitely axiomatizable}$ (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.

Replaces 'Finite axiomatizability for profinite groups I: group theory', adding significant additional material. New version has better proofs of bi-intepretability