A pro-p group G is a PF-group if it has central series of closed subgroups {N i } i∈N with trivial intersection satisfying N

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Abstract. According to Lazard, every p-adic Lie group contains an open pro-p subgroup which is saturable. This can be regarded as the starting point of p-adic Lie theory, as one can naturally associate to every saturable pro-p group G a Lie lattice LðGÞ over the p-adic integers.Essential features of saturable pro-p groups include that they are torsion-free and p-adic analytic. In the present paper we prove a converse result in small dimensions: every torsion-free p-adic analytic pro-p group of dimension less than p is saturable.This leads to useful consequences and interesting questions. For instance, we give an e¤ec-tive classification of 3-dimensional soluble torsion-free p-adic analytic pro-p groups for p > 3. Our approach via Lie theory is comparable with the use of Lazard's correspondence in the classification of finite p-groups of small order.

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