The admissibility of \(M_{11}\) over number fields
type: Item
instance of: scholarly article
title: The admissibility of \(M_{11}\) over number fields
zbMATH Open document ID: 1453.12005
author: Joachim König
author: Danny Neftin
published in: Journal of Pure and Applied Algebra
publication date: 2018-05-02T00:00:00Z
review text: Let \(k\) be a field, \(D\) a central division \(k\)-algebra and \(G\) a finite group. We say that \(D\) is a \(G\)-crossed product if it admits a maximal subfield \(L\) that is a Galois extension of \(k\) with a Galois group \(\mathcal{G}(L/k)\) isomorphic to \(G\). A Galois extension \(L/k\) with \(\mathcal{G}(L/k) \cong G\) is called \(k\)-adequate, if \(L\) is a maximal subfield of a \(G\)-crossed product central division \(k\)-algebra. The problem of finding the \(k\)-admissible finite groups, i.e. those groups \(G\) for which there exists a \(G\)-crossed product division algebra central over \(k\), is a classical problem in the theory of central division algebras over special fields. This problem originates from the study of \(\mathbb{Q}\)-admissibility initiated by Schacher who proved that every \(\mathbb{Q}\)-admissible group has metacyclic Sylow subgroups (see [\textit{M. Schacher}, J. Algebra 9, 451--477 (1968; Zbl 0174.34103)]. Conversely, it has been proved by \textit{J. Sonn} [J. Algebra 84, 411--419 (1983; Zbl 0524.16007)] that finite solvable groups whose Sylow subgroups are metacyclic are necessarily \(\mathbb{Q}\)-admissible. Moreover, by Sonn's theorem, every group \(H\) of the considered type is strongly \(\mathbb{Q}\)-admissible, in the sense that, for each \(n \in \mathbb{N}\), there is a \(\mathbb{Q}\)-adequate Galois extension \(L/\mathbb{Q}\) with \(\mathcal{G}(L/\mathbb{Q}) \cong H\) and \(L \cap \mathbb{Q}(\mu_n) = \mathbb{Q}\), where \(\mu_n\) denotes the group of \(n\)-th roots of unity in a separable closure of \(\mathbb{Q}\). Generally, the question of whether finite groups with metacyclic Sylow subgroups are \(\mathbb{Q}\)-admissible remains open. It is known that the answer to this question reduces to proving the strong \(\mathbb{Q}\)-admissibility of the groups in an explicit list of five families (see [\textit{D. Chillag} and \textit{J. Sonn}, Isr. J. Math. 40, 307--323 (1981; Zbl 0496.20027)]. The last family in this list is the Ma…
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 6865843
zbMATH Keywords: \(G\)-crossed product
zbMATH Keywords: \(k\)-admissible group
zbMATH Keywords: adequate extension
zbMATH Keywords: \(\mathbb{Q}\)-admissibility conjecture
zbMATH Keywords: Mathieu group \(M_{11}\)
MaRDI profile type: Publication
full work available at URL: j.jpaa.2017.09.022
OpenAlex ID: W2757184328
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DOI: 10.1016/J.JPAA.2017.09.022
description: scientific article; zbMATH DE number 6865843