Finite \(p\)-groups all of whose proper subgroups have small derived subgroups.
type: Item
instance of: scholarly article
title: Finite \(p\)-groups all of whose proper subgroups have small derived subgroups.
zbMATH Open document ID: 1202.20022
author: Junqiang Zhang
author: Xianhua Li
published in: Science China. Mathematics
publication date: 2010-08-23T00:00:00Z
review text: Suppose that all proper subgroups of a group \(G\) have derived subgroups of order \(\leq p\) (the authors call such groups \(D_1\)-groups). It is known that \(p\)-groups all of whose subgroups of index \(p^2\) are Abelian, are D\(_1\)-groups. The main theorem classifies \(D_1\)-groups. If \(G\) is a \(D_1\)-group, then \(|G'|\leq p^3\), \(d(G)\leq p^3\), \(cl(G)\leq 3\) (so that, if \(p>3\), then \(G\) is regular).
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D25
zbMATH DE Number: 5774126
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: derived subgroup
zbMATH Keywords: minimal non-Abelian \(p\)-groups
zbMATH Keywords: \(D_1\)-groups
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: s11425-010-0010-1
OpenAlex ID: W2045987889
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DOI: 10.1007/S11425-010-0010-1
description: scientific article; zbMATH DE number 5774126