On finite \(p\)-groups with few nonabelian subgroups of order \(p^p\) and exponent \(p\).
type: Item
instance of: scholarly article
title: On finite \(p\)-groups with few nonabelian subgroups of order \(p^p\) and exponent \(p\).
zbMATH Open document ID: 1208.20020
published in: Israel Journal of Mathematics
publication date: 2010-12-06T00:00:00Z
review text: The finite \(p\)-groups that are minimal nonabelian have been classified by \textit{L.~Rédei} [Comment. Math. Helv. 20, 225-264 (1947; Zbl 0035.01503)]. If \(G\) is a finite, nonabelian \(p\)-group, then its minimal nonabelian subgroups generate \(G\). The main result of this paper (Theorem~1.1) deals with a generalization of this fact. Let \(G\) be a finite, nonabelian group. Assume \(G\) is generated by elements of order at most \(p^k\), for some \(k\). Then \(G\) is generated by those minimal nonabelian subgroups, that are generated by elements of order at most \(p^{k+1}\). As an application, counting theorems for the number of certain subgroups of maximal class in a \(p\)-group are obtained.
reviewed by: Andrea Caranti
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 5823087
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: minimal nonabelian groups
zbMATH Keywords: \(p\)-groups of maximal class
zbMATH Keywords: numbers of subgroups
author: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: s11856-010-0078-x
OpenAlex ID: W2008031680
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DOI: 10.1007/S11856-010-0078-X
description: scientific article; zbMATH DE number 5823087