On Galois cohomology and realizability of 2-groups as Galois groups
type: Item
instance of: scholarly article
title: On Galois cohomology and realizability of 2-groups as Galois groups
zbMATH Open document ID: 1256.12004
author: Ivo M. Michailov
published in: Central European Journal of Mathematics
publication date: 2011-05-27T00:00:00Z
review text: The paper under review continues the presentation of the author's research on the problem of finding whether a given finite \(p\)-group \(G\) is realizable as a Galois group over an arbitrary field \(F\) of characteristic different from \(p\) (the solution to the same problem in characteristic \(p\) is well-known consequence of the projectivity of the Galois group of the maximal \(p\)-extension). The author concentrates on the case where \(p = 2\), \(G\) is of order \(2 ^ n\), for an integer \(n \geq 4\), and \(G\) possesses a cyclic group of order \(2 ^ {n-2}\). Assuming also that \(F\) contains a primitive root of unity of degree \(2 ^ {n-3}\), \(G\) is nonabelian and \(G\) does not decompose into a proper group product, he finds necessary and sufficient conditions for realizability of \(G\) over \(F\), when \(G\) lies in \(14\) of \(22\) possible isomorphism classes. The proof of the stated result relies on \textit{Y. Ninomiya}'s classification of nonabelian finite groups of order \(p ^ n\) and exponent \(p ^ {n-2}\) (see [Math. J. Okayama Univ. 36, 1--21 (1994; Zbl 0838.20017)]). It is obtained by considering several special cases. They are settled by applying some criteria for solving embedding problems involving \(p\)-groups (one of them obtained by \textit{I. Kiming} [Can J. Math. 42, No. 5, 825--855 (1990; Zbl 0725.12004)]), which are of independent interest. This is combined with a method (worked out in the paper), applicable to \(2\)-groups, that is based on the corestriction mapping of Brauer groups.
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 12G05
zbMATH DE Number: 5900628
zbMATH Keywords: Embedding problem
zbMATH Keywords: Galois extension
zbMATH Keywords: quaternion algebra
zbMATH Keywords: obstruction
zbMATH Keywords: corestriction
zbMATH Keywords: \(p\)-groups
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: s11533-011-0004-4
OpenAlex ID: W2129051423
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DOI: 10.2478/S11533-011-0004-4
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description: scientific article; zbMATH DE number 5900628