\(p\)-groups with maximal elementary Abelian subgroups of rank 2.
type: Item
instance of: scholarly article
title: \(p\)-groups with maximal elementary Abelian subgroups of rank 2.
zbMATH Open document ID: 1194.20017
author: George Glauberman
author: Nadia Mazza
published in: Journal of Algebra
publication date: 2010-03-29T00:00:00Z
review text: Let \(G\) be a finite \(p\)-group with the prime \(p\) odd. The main result of the paper is that if the group \(G\) possesses a maximal elementary Abelian subgroup of rank 2 then the group \(G\) has rank at most \(p\), answering a question raised by the second author [in J. Algebra 320, No. 12, 4242-4248 (2008; Zbl 1165.20016)]. For the 2-group case it is shown that the group, with a maximal elementary Abelian subgroup of rank 2, should have rank at most 4, and an example is given that indeed there is such a 2-group of rank 4. Considering the poset of elementary Abelian subgroups ordered by inclusion, the main result has the consequence that if a finite group \(H\) has \(p\)-rank greater than \(p\) then the group of endotrivial \(kH\)-modules with \(k\) a field of characteristic \(p\) has torsion-free rank 1, applying the classification of endotrivial modules settled by \textit{J. F. Carlson} and \textit{J. Thévenaz} [Invent. Math. 158, No. 2, 389-411 (2004; Zbl 1110.20004)]. In connection with the class-breadth conjecture, recently studied by \textit{B. Eick, M. F. Newman, E. A. O'Brien}, [J. Algebra 300, No. 1, 384-393 (2006; Zbl 1108.20014)], the main result has the consequence that if for a finite \(p\)-group the poset of elementary Abelian subgroups has more than one component then the class-breadth conjecture is satisfied, that is, the group is of class not exceeding the class-breadth by more than one.
reviewed by: János Kurdics
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 19A22
zbMATH DE Number: 5687634
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: elementary Abelian \(p\)-subgroups
zbMATH Keywords: endotrivial modules
zbMATH Keywords: class-breadth conjecture
zbMATH Keywords: posets of Abelian subgroups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.10.015
OpenAlex ID: W2012664236
cites work: Limits of Abelian subgroups of finite \(p\)-groups
cites work: Groups of prime power order. Vol. 1.
cites work: Groups of prime-power order having an Abelian centralizer of type (r,1)
cites work: Maximal elementary abelian subgroups of rank 2
cites work: Endotrivial modules for finite groups of Lie type
cites work: The classification of endo-trivial modules.
cites work: The class-breadth conjecture revisited.
cites work: Solvability of groups of odd order
cites work: Q5577154
cites work: Q4857335
cites work: Q3253828
cites work: A Contribution to the Theory of Groups of Prime-Power Order
cites work: Q3698995
cites work: On Subgroups of p -Groups Having Soft Subgroups
cites work: Endliche Gruppen I
cites work: Counting abelian subgroups of p-groups. A projective approach
cites work: Q4377749
cites work: The Breadth and the Class of a Finite P -Group
cites work: On 2-Groups With No Normal Abelian Subgroups of Rank 3, and Their Occurrence as Sylow 2-Subgroups of Finite Simple Groups
cites work: Q3256242
DOI: 10.1016/J.JALGEBRA.2009.10.015
description: scientific article; zbMATH DE number 5687634