On p-saturable groups

groups, respectively); see [, § 4] for an overview and [, Chapter 4] for details. While the general interplay between subalgebras and subgroups is subtle, we note the following fact.…”

Section: Orbits and Conjugacy Classes Of Linear Groupsmentioning

confidence: 99%

The average size of the kernel of a matrix and orbits of linear groups

Rossmann

2018

Proc. London Math. Soc.

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Let frakturO be a compact discrete valuation ring of characteristic 0. Given a module M of matrices over frakturO, we study the generating function encoding the average sizes of the kernels of the elements of M over finite quotients of frakturO. We prove rationality and establish fundamental properties of these generating functions and determine them explicitly for various natural families of modules M. Using p‐adic Lie theory, we then show that special cases of these generating functions enumerate orbits and conjugacy classes of suitable linear pro‐p groups.

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“…Let R be the soluble radical of the saturable Lie lattice L. By Proposition 5.1, R is isolated and PF-embedded in L. In particular, R is saturable. According to [5,Theorem 4.5 and Corollary 4.7], the subset R forms a soluble PF-embedded subgroup of the saturable group G. Clearly, R is also isolated as a group.…”

Section: Furthermore If L Is a Saturable Z P -Lie Lattice And M Is Amentioning

confidence: 99%

Analytic pro-p groups of small dimensions

González-Sánchez¹,

Klopsch²

2009

Journal of Group Theory

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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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On p-saturable groups

Abstract: 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

Cited by 19 publications

References 6 publications

Omega subgroups of pro-p groups

Omega subgroups of pro-p groups

The average size of the kernel of a matrix and orbits of linear groups

Analytic pro-p groups of small dimensions

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