A note on Hamiltonian \(2\)-groups
type: Item
instance of: scholarly article
title: A note on Hamiltonian \(2\)-groups
zbMATH Open document ID: 0799.20023
author: R. A. Bryce
author: John Cossey
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1992-09-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108231
review text: A nonabelian finite group is called a Hamiltonian group if every subgroup is normal. A Hamiltonian 2-group is a direct product of a quaternion group of order 8 and an elementary abelian 2-group. Denote by \(H_ n\) a Hamiltonian group of order \(2^{n + 2}\). The authors study how a Hamiltonian 2-group occurs as a normal section of a 2-group. Let \(\beta(G)\) be the subgroup of \(G\) generated by \([S,G]\), where \(S\) runs over all non-normal subgroups of \(G\). It is shown that if \(G/\beta(G)\) is Hamiltonian, then \(\beta(G) = 1\). Another result is that for \(c \geq 2^ m\) there is a 2-group whose \(m\)-th derived subgroup is isomorphic to \(H_ c\). The authors also obtain the upper bound of the derived length of a 2-group which contains a normal Hamiltonian subgroup.
reviewed by: Hiromichi Yamada
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 60487
zbMATH Keywords: finite group
zbMATH Keywords: Hamiltonian 2-group
zbMATH Keywords: quaternion group
zbMATH Keywords: elementary abelian 2-group
zbMATH Keywords: normal section
zbMATH Keywords: derived subgroup
zbMATH Keywords: derived length
MaRDI profile type: Publication
cites work: Endliche Gruppen I
description: scientific article; zbMATH DE number 60487