On Hua-Tuan's conjecture. II.
type: Item
instance of: scholarly article
title: On Hua-Tuan's conjecture. II.
zbMATH Open document ID: 1223.20010
author: Qinhai Zhang
author: Haipeng Qu
published in: Science China. Mathematics
publication date: 2011-07-01T00:00:00Z
review text: According to Kulakoff's theorem, the number of nontrivial subgroups in a noncyclic \(p\)-group \(G\) is equal to \(1+p+kp^2\) for some nonnegative integer. More than 60 years ago Hua and Tuan conjectured that \(k\in\{0,1,2\}\). The authors in this and the previous paper [part I, ibid. 52, No. 2, 389-393 (2009; Zbl 1181.20019)] constructed \(p\)-groups with \(k=3\). They conjectured that \(k\in\{0,1,2,3\}\). The paper is devoted to checking this conjecture for a number of small \(p\)-groups. (I think that \(k\) can be equal to every member of the set \(\{0,1,2,\dots,p-1\}\).)
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 5916771
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: numbers of subgroups
zbMATH Keywords: Hua-Tuan conjecture
zbMATH Keywords: \(p\)-groups of maximal class
zbMATH Keywords: enumerations of subgroups
zbMATH Keywords: Magma package
reviewed by: Yakov Berkovich
describes a project that uses: Magma
MaRDI profile type: Publication
Wikidata QID: Q122977724
full work available at URL: s11425-010-4136-y
OpenAlex ID: W2056200404
cites work: On the Number of Subgroups of Given Order in a Finite p-Group of Exponent p
cites work: Groups of prime power order. Vol. 1.
cites work: Generalizations of Certain Elementary Theorems on p -Groups
cites work: The Magma algebra system. I: The user language
cites work: Endliche Gruppen I
cites work: Finite \(p\)-groups in which the number of subgroups of possible order is less than or equal to \(p^3\).
cites work: Q5798835
cites work: Q3786631
cites work: A classification of some regular \(p\)-groups and its applications.
cites work: On Hua-Tuan's conjecture.
DOI: 10.1007/S11425-010-4136-Y
description: scientific article; zbMATH DE number 5916771