Isoclinic Galois groups
type: Item
instance of: scholarly article
title: Isoclinic Galois groups
zbMATH Open document ID: 1327.12002
author: D. Kharzeev
published in: Beiträge zur Algebra und Geometrie
publication date: 2015-10-16T00:00:00Z
review text: Two groups \(G\) and \(H\) are called isoclinic if there exist isomorphisms \(\theta: G/Z(G)\to H/Z(H)\) and \(\phi:G'\to H'\) such that \(\phi([x,y])=[x',y']\), where \(x'Z(H)=\theta(xZ(G))\) and \(y'Z(H)=\theta(yZ(G))\). (Here \(Z(G)\) is the centre of \(G\), and \(G'\) is the commutator subgroup of \(G\), generated by all commutators \([x,y]=x^{-1}y^{-1}xy\).) In this paper Schmid gives a different description for stem groups. A group \(X\) is a stem group provided \(Z(X)\subseteq X'\). The author considers the embedding problem in Galois theory related to a central stem extension \(1\to Z\to X\to G\to 1\) of a finite group \(G\). As a corollary of the main result Schmid shows that if \(X\) is realizable as a Galois group over any number field \(k\), then all groups in the isoclinism class of \(X\) are also realizable as Galois groups over \(k\). The other groups appear as Galois groups of subfields of the compositum of certain abelian extensions of \(k\), whose existence can be determined by class field theory. The obtained results can be used in the study of the realizability of \(p\)-groups as Galois groups over number fields.
reviewed by: Ivo M. Michailov
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 6495600
zbMATH Keywords: isoclinism
zbMATH Keywords: Schur multiplier
zbMATH Keywords: Galois theory
zbMATH Keywords: embedding problem
zbMATH Keywords: inverse problem
MaRDI profile type: Publication
full work available at URL: s13366-014-0234-2
OpenAlex ID: W2051345143
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DOI: 10.1007/S13366-014-0234-2
description: scientific article; zbMATH DE number 6495600