More on core-2 2-groups
type: Item
instance of: scholarly article
title: More on core-2 2-groups
zbMATH Open document ID: 1370.20017
author: Bettina Wilkens
published in: Journal of Group Theory
publication date: 2017-03-14T00:00:00Z
review text: Let \(G\) be a group and \(H\) be a subgroup. Denote by \(H_G = \cap_{g \in G} H^g\) the core of \(H\). A finite group \(G\) is called a core-\(n\) group if \(|H : H_G| \leq n\) for all subgroups \(H\) of \(G\). This paper is devoted to \(2\)-groups \(G\), which are core-\(2\) groups. The main result says that any such group contains an abelian subgroup \(A\) of index at most 4, which contains \(\Phi (G)\). As a corollary one gets that any core-\(2\) group contains an abelian subgroup of index at most eight. This bound is best possible as \(G \cong D_{16} \ast Q_8\) shows. The corresponding result for odd \(p\), i.e., \(p\)-groups \(G\) (\(p\) odd) which are core-\(p\) have a normal abelian subgroup of index \(p^2\), was proved by \textit{G. Cutolo} et al. [J. Algebra 188, No. 2, 701--719 (1997; Zbl 0878.20014)].
reviewed by: Gernot Stroth
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 6694617
zbMATH Keywords: \(2\)-groups
MaRDI profile type: Publication
full work available at URL: jgth-2016-0035
OpenAlex ID: W2507923434
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DOI: 10.1515/JGTH-2016-0035
description: scientific article; zbMATH DE number 6694617