Projective Groups and Frattini Covers

Abstract

In this chapter we define a projective group as a profinite group 𝐺 for which every embedding problem is weakly solvable. By Gruenberg’s theorem (Lemma 25.3.2), it suffices to weakly solve only finite embedding problems. This leads to the second characterization of projective groups as those profinite groups which are isomorphic to closed subgroups of free profinite groups (Corollary 25.4.6).