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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W.

Managing Editor: Wilson, John S. / Khukhro, Evgenii I. / Kramer, Linus

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Source Normalized Impact per Paper (SNIP) 2018: 1.047

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Volume 12, Issue 5


Analytic pro-p groups of small dimensions

Jon González-Sánchez
  • Departmento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avda. de los Castros, E-39071 Santander, Spain. E-mail:
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/ Benjamin Klopsch
  • Department of Mathematics, Royal Holloway, University of London, Egham TW20 0EX, United Kingdom. E-mail:
  • Other articles by this author:
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Published Online: 2009-04-17 | DOI: https://doi.org/10.1515/JGT.2009.006


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.

About the article

Received: 2007-11-15

Revised: 2008-06-18

Published Online: 2009-04-17

Published in Print: 2009-09-01

Citation Information: Journal of Group Theory, Volume 12, Issue 5, Pages 711–734, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/JGT.2009.006.

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