Abstract

AbstractLetpbe a prime. A pro-pgroupGis said to be 1-smooth if it can be endowed with a continuous representation$\theta \colon G\to \mathrm {GL}_1(\mathbb {Z}_p)$such that every open subgroupHofG, together with the restriction$\theta \vert _H$, satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-pgroup contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-pGalois groups of fields, and that if a 1-smooth pro-pgroup is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-pGalois groups of fields. Finally, we ask whether 1-smooth pro-pgroups satisfy a “Tits’ alternative.”

Highlights

  • Throughout the paper p will denote a prime number, and K a field containing a root of unity of order p

  • One of the obstructions for the realization of a pro-p group as maximal pro-p Galois group for some field K is given by the Artin–Scherier theorem: the only finite group realizable as GK(p) is the cyclic group of order 2

  • The now-called Norm Residue Theorem implies that the Z/p-cohomology algebra of a maximal pro-p Galois group GK(p)

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Summary

Introduction

Throughout the paper p will denote a prime number, and K a field containing a root of unity of order p. One of the obstructions for the realization of a pro-p group as maximal pro-p Galois group for some field K is given by the Artin–Scherier theorem: the only finite group realizable as GK(p) is the cyclic group of order 2 (cf [1]). H●(GK(p), Z/p) ∶= ⊕ Hn(GK(p), Z/p), n≥0 with Z/p a trivial GK(p)-module and endowed with the cup-product, is a quadratic algebra: i.e., all its elements of positive degree are combinations of products of elements of degree 1, and its defining relations are homogeneous relations of degree 2 (see Section 2.3) From this property one may recover the Artin-Schreier obstruction (see, e.g., [17, Section 2]).

Quadrelli
Cyclotomic pro-p pairs
Kummerian pro-p pairs
The Galois case
Bloch–Kato and the Smoothness Conjecture
Powerful pro-p groups
Solvable pro-p groups and maximal pro-p Galois groups
A Tits’ alternative for 1-smooth pro-p groups

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