On p-saturable groups

A pro-p group G is a PF-group if it has central series of closed subgroups with trivial intersection satisfying N1=G and . In this paper, we prove that a finitely generated pro-p group G is a p-saturable group, in the sense of Lazard, if and only if it is a torsion free PF-group. Using this characterization, we study certain families of subgroups of p-saturable groups. For example, we prove that any normal subgroup of a p-saturable group contained in the Frattini is again p-saturable.