Finitely generated groups and countably based groups

Abstract We are concerned here with profinite groups whose topology has a countable base for its open sets. They may be characterized in various ways-they are, for example, the groups which are inverse limits of inverse systems of finite groups indexed by N. These groups, the countably based profinite groups, play a role similar to the role played by countable groups in abstract group theory. Examples are the profinite groups which are topologically finitely generated, and every countably based profinite group arises as a closed subgroup of a finitely generated profinite group. Indeed, for many classes C of finite groups, there exists a finitely generated pro-C group in which all countably based pro-C groups can be embedded.