Finite \(p\)-groups with many minimal nonabelian subgroups.
type: Item
instance of: scholarly article
title: Finite \(p\)-groups with many minimal nonabelian subgroups.
zbMATH Open document ID: 1262.20022
author: Zvonimir Janko
published in: Journal of Algebra
publication date: 2012-11-09T00:00:00Z
review text: The author in a series of papers, [such as Glas. Mat., III. Ser. 46, No. 2, 351-356 (2011; Zbl 1244.20014)], investigates the structure of finite \(p\)-groups. Let \(G\) be a nonabelian not minimal nonabelian finite \(p\)-group. Consider the following two properties: (*) any nonabelian subgroup \(H\) of \(G\) with an Abelian maximal subgroup is minimal nonabelian; (**) whenever \(X\) and \(Y\) are two minimal nonabelian subgroups, the elements \(x\in X\setminus Y\) and \(y\in Y\setminus X\) generate a minimal nonabelian subgroup. The author describes completely the groups \(G\) satisfying one of these properties. As to the latter one, if it is satisfied then the former one is also satisfied, \(p=2\), \(G\) is special of order 64 and isomorphic to the 2-Sylow subgroup of the simple group \(Sz(8)\). This answers problem 2331 in the forthcoming book by \textit{Y. Berkovich} and the author [Groups of prime power order. Vol. 4. Berlin: Walter de Gruyter (in preparation)].
reviewed by: János Kurdics
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 6104538
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: minimal nonabelian \(p\)-groups
zbMATH Keywords: metacyclic \(p\)-groups
zbMATH Keywords: Suzuki 2-groups
zbMATH Keywords: Abelian maximal subgroups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.02.014
OpenAlex ID: W2020443179
cites work: Groups of prime power order. Vol. 1.
cites work: Groups of prime power order. Vol. 2.
cites work: Groups of prime power order. Vol. 3.
DOI: 10.1016/J.JALGEBRA.2012.02.014
description: scientific article; zbMATH DE number 6104538