Finite \(p\)-groups with a minimal non-Abelian subgroup of index \(p\). I.
type: Item
instance of: scholarly article
title: Finite \(p\)-groups with a minimal non-Abelian subgroup of index \(p\). I.
zbMATH Open document ID: 1262.20023
author: Haipeng Qu
author: Sushan Yang
author: Lijian An
author: Mingyao Xu
published in: Journal of Algebra
publication date: 2012-11-09T00:00:00Z
review text: Let \(G\) be a nonabelian \(p\)-group with \(p\) odd. \(G\) is called an \(A_k\)-group if each of its subgroups of index \(p^k\) is Abelian but this does not hold for subgroups of index \(p^{k-1}\). The problem of classifying \(A_2\)-groups has been solved recently by \textit{Q. Zhang, X. Sun, L. An} and \textit{M. Xu} [in Algebra Colloq. 15, No. 1, 167-180 (2008; Zbl 1153.20018)]. This is the first part of a series of papers solving the following problem due to Berkovich: Classify groups \(G\) containing an \(A_1\)-subgroup of index \(p\). The question is split into two parts, in the present paper the simpler one is solved as follows: the group \(G\) contains a unique \(A_1\)-subgroup of index \(p\) if and only if either it is such a group of order \(3^5\); or it is of order greater than \(p^5\), it has no Abelian maximal subgroup and has a maximal section which is a 3-group of maximal class and has an Abelian maximal subgroup. Moreover, isomorphism classes of these 3-groups \(G\) are completely described by generators and relations.
reviewed by: János Kurdics
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 6104621
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: minimal non-Abelian \(p\)-groups
zbMATH Keywords: metabelian \(p\)-groups
zbMATH Keywords: regular \(p\)-groups
zbMATH Keywords: \(p\)-groups of maximal class
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.03.007
OpenAlex ID: W2046306908
cites work: Finite \(p\)-groups with a minimal non-abelian subgroup of index \(p\). II.
cites work: Finite p-groups with a minimal non-abelian subgroup of index p (IV)
cites work: Q5480910
cites work: Groups of prime power order. Vol. 1.
cites work: Groups of prime power order. Vol. 2.
cites work: On a special class of \(p\)-groups
cites work: Generalizations of Certain Elementary Theorems on p -Groups
cites work: Q3973708
cites work: Endliche Gruppen I
cites work: Über das Produkt von paarweise vertauschbaren zyklischen Gruppen
cites work: Q5631292
cites work: Application of modules over a dyad for the classification of finite \(p\)-groups possessing an Abelian subgroup of index \(p\) and of pairs of mutually annihilating operators
cites work: Finite \(p\)-groups with a minimal non-abelian subgroup of index \(p\). III.
cites work: Finite \(p\)-groups with a minimal non-Abelian subgroup of index \(p\). V.
cites work: Q2648167
cites work: Q5664075
cites work: Q5799445
cites work: A theorem on metabelian p-groups and some consequences
cites work: Finite \(p\)-groups all of whose non-Abelian proper subgroups are generated by two elements.
cites work: Finite p-Groups All of Whose Subgroups of Index p2 Are Abelian
DOI: 10.1016/J.JALGEBRA.2012.03.007
description: scientific article; zbMATH DE number 6104621