  We give a complete characterization of countable primitive groups in several
settings including linear groups, subgroups of mapping class groups, groups
acting minimally on trees and convergence groups. The latter category includes
as a special case Kleinian groups as well as subgroups of word hyperbolic
groups. As an application we calculate the Frattini subgroup in many of these
settings, often generalizing results that were only known for finitely
generated groups. In particular, we answer a question of G. Higman and B.H.
Neumann on the Frattini group of an amalgamated product.
