Finite axiomatizability for profinite groups

A group is finitely axiomatizable (FA) in a class if it can be determined up to isomorphism within 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, or in the class of pro- groups for some prime . Both algebraic and model-theoretic methods are developed for the purpose. Reasons why certain groups cannot be FA are also discussed