On exceptional \(pq\)-groups
type: Item
instance of: scholarly article
title: On exceptional \(pq\)-groups
zbMATH Open document ID: 1275.11146
author: Jerzy Browkin
author: Kejian Xu
published in: Science China. Mathematics
publication date: 2013-01-28T00:00:00Z
review text: Let \(F/k\) be a normal extension of number fields with Galois group \(G\). If in the Brauer-Kuroda relation of the Dedekind zeta functions of intermediate fields of \(F/k\) the zeta function of \(F\) does not appear (this only depends on \(G\)), then the group \(G\) is said to be exceptional (see \textit{J. Browkin} et al. [Bull. Pol. Acad. Sci., Math. 59, No. 3, 207--214 (2011; Zbl 1244.20013)]). In this article, the authors investigate the exceptional groups whose orders are products of powers of two primes \(p\) and \(q\) such that \(G\) has unique subgroups of orders \(p\) and \(q\). More precisely, let \(G\) be as above, let \(N_d(G)\) denote the number of cyclic subgroups of \(G\) with order \(d\), and set \(N(G) = (N_p(G),N_q(G), N_{pq}(G))\). Thm.~4.1 shows that if \(N_p(G) = N_q(G) = 1\), then the Sylow subgroups of \(G\) are cyclic or generalized quaternion. This is used in Thm.~4.5. to classify all such groups \(G\) with \(N(G) = (1,1,1)\).
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 20D99
zbMATH DE Number: 6131125
zbMATH Keywords: Brauer-Kuroda relation
zbMATH Keywords: exceptional group
zbMATH Keywords: class number relations
zbMATH Keywords: Dedekind zeta function
MaRDI profile type: Publication
full work available at URL: s11425-012-4460-5
OpenAlex ID: W2041749679
cites work: Higher Regulators, Algebraic 𝐾-Theory, and Zeta Functions of Elliptic Curves
cites work: Beziehungen zwischen Klassenzahlen von Teilkörpern eines galoisschen Körpers. Herrn Professor Dr. Erhard Schmidt in dankbarer Verehrung zum 75. Geburtstag
cites work: On Exceptions in the Brauer–Kuroda Relations
cites work: Q3708977
cites work: Artin-root numbers and normal integral bases for quaternion fields
cites work: Q5843509
cites work: Über die Klassenzahlen algebraischer Zahlkörper
cites work: Q4770462
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: Q3062972
DOI: 10.1007/S11425-012-4460-5
description: scientific article; zbMATH DE number 6131125