Abstract
AbstractLet p be a prime. A pro-p group G is 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 subgroup H of G, together with the restriction $\theta \vert _H$ , satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-p group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-p Galois groups of fields, and that if a 1-smooth pro-p group is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-p Galois groups of fields. Finally, we ask whether 1-smooth pro-p groups 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)
Summary
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]).
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