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Publicly Available Published by De Gruyter April 17, 2009

Analytic pro-p groups of small dimensions

  • Jon González-Sánchez and Benjamin Klopsch
From the journal Journal of Group Theory

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 L(G) 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 effective 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.

Received: 2007-11-15
Revised: 2008-06-18
Published Online: 2009-04-17
Published in Print: 2009-September

© de Gruyter 2009

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