Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter July 18, 2017

On the structure of virtually nilpotent compact p-adic analytic groups

William Woods
From the journal Journal of Group Theory


Let G be a compact p-adic analytic group. We recall the well-understood finite radical Δ+ and FC-centre Δ, and introduce a p-adic analogue of Roseblade’s subgroup nio(G), the unique largest orbitally sound open normal subgroup of G. Further, when G is nilpotent-by-finite, we introduce the finite-by-(nilpotent p-valuable) radical 𝐅𝐍p(G), an open characteristic subgroup of G contained in nio(G). By relating the already well-known theory of isolators with Lazard’s notion of p-saturations, we introduce the isolated lower central (resp. isolated derived) series of a nilpotent (resp. soluble) p-valuable group, and use this to study the conjugation action of nio(G) on 𝐅𝐍p(G). We emerge with a structure theorem for G,


in which the various quotients of this series of groups are well understood. This sheds light on the ideal structure of the Iwasawa algebras (i.e. the completed group rings kG) of such groups, and will be used in future work to study the prime ideals of these rings.

Communicated by John S. Wilson


[1] K. Ardakov, Localisation at augmentation ideals in Iwasawa algebras, Glasg. Math. J. 48 (2006), no. 2, 251–267. 10.1017/S0017089506003041Search in Google Scholar

[2] K. Ardakov, Prime ideals in nilpotent Iwasawa algebras, Invent. Math. 190 (2012), no. 2, 439–503. 10.1007/s00222-012-0385-4Search in Google Scholar

[3] K. A. K. Brown, Primeness, semiprimeness and localisation in Iwasawa algebras, Trans. Amer. Math. Soc. 359 (2007), 1499–1515. 10.1090/S0002-9947-06-04153-5Search in Google Scholar

[4] J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic Pro-p Groups, Cambridge University Press, Cambridge, 1999. 10.1017/CBO9780511470882Search in Google Scholar

[5] M. Lazard, Groupes analytiques p-adiques, Publ. Math. Inst. Hautes Études Sci. 26 (1965), 5–219. 10.1007/BF02684303Search in Google Scholar

[6] M. L. E. Letzter, Polycyclic-by-finite group algebras are catenary, Math. Res. Lett. 6 (1999), 183–194. 10.4310/MRL.1999.v6.n2.a6Search in Google Scholar

[7] J. McConnell and J. Robson, Noncommutative Noetherian Rings, American Mathematical Society, Providence, 2001. 10.1090/gsm/030Search in Google Scholar

[8] J. Nelson, Localisation and Euler characteristics of soluble Iwasawa algebras, Ph.D. thesis, University of Cambridge, 2013. Search in Google Scholar

[9] A. Neumann, Completed group algebras without zero divisors, Arch. Math. 51 (1988), 496–499. 10.1007/BF01261969Search in Google Scholar

[10] D. S. Passman, Algebraic Structure of Group Rings, John Wiley & Sons, New York, 1977. Search in Google Scholar

[11] D. S. Passman, Infinite Crossed Products, Academic Press, Boston, 1989. Search in Google Scholar

[12] D. J. Robinson, A Course in the Theory of Groups, Springer, New York, 1996. 10.1007/978-1-4419-8594-1Search in Google Scholar

[13] J. Roseblade, Prime ideals in group rings of polycyclic groups, Proc. Lond. Math. Soc. (3) 36 (1978), 385–447. 10.1112/plms/s3-36.3.385Search in Google Scholar

[14] D. Segal, Polycyclic Groups, Cambridge University Press, Cambridge, 2005. Search in Google Scholar

Received: 2017-3-28
Published Online: 2017-7-18
Published in Print: 2018-1-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Scroll Up Arrow