Large subgroups of small class in finite \(p\)-groups.
type: Item
instance of: scholarly article
title: Large subgroups of small class in finite \(p\)-groups.
zbMATH Open document ID: 1048.20004
author: George Glauberman
published in: Journal of Algebra
publication date: 2004-03-15T00:00:00Z
review text: Abelian subgroups play a key role in the theory and applications of finite \(p\)-groups. For example, J. G. Thompson strengthened his celebrated theorem, the Frobenius kernel of a Frobenius group is nilpotent, by using the Abelian subgroups of maximal rank in a Sylow subgroup of a finite group [\textit{J. G. Thompson}, J. Algebra 1, 43-46 (1964; Zbl 0119.26802)]. One may ask whether one of these subgroups is normal in the Sylow subgroup. More generally, the following very interesting questions are posed in the present paper: Question 1. Suppose \(A\) is an elementary Abelian subgroup of order \(p^n\) in a finite \(p\)-group \(P\). Does \(P\) possess a normal elementary Abelian subgroup of order \(p^n\)? Question 2. Suppose \(A\) is an Abelian subgroup of order \(p^n\) in a finite \(p\)-group \(P\). Does \(P\) possess a normal Abelian subgroup of order \(p^n\)? In general, the answers to the above questions are all ``no'', but affirmative under certain conditions. In a remarkable paper [\textit{J. L. Alperin} and \textit{G. Glauberman}, J. Algebra 203, No. 2, 533-566 (1998; Zbl 0964.20011)], using an idea from algebraic group theory, the authors provided affirmative answers to some special cases of Question 1 and obtained some related results for \(p\)-groups and nilpotent Lie algebras over fields of characteristic \(p\) under the restriction on \(A\) being elementary Abelian. In this paper, the author removes the restriction and obtains analogues for Question 2 of the results about Question 1 by extending the application of algebraic group theory. He also obtains generalizations to subgroups of small nilpotence class, not necessarily Abelian, and to nilpotent Lie algebras over local rings. Five rather complicated theorems are proved in the present paper.
reviewed by: Shi Wujie
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F40
zbMATH DE Number: 2057667
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: nilpotent Lie algebras
zbMATH Keywords: unipotent automorphisms
zbMATH Keywords: normal elementary Abelian subgroups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2002.11.003
OpenAlex ID: W2002408953
cites work: Limits of Abelian subgroups of finite \(p\)-groups
cites work: On a nonregular \(p\)-group
cites work: A Measuring Argument for Finite Groups
cites work: Large Abelian subgroups of finite \(p\)-groups
cites work: Q4143433
cites work: Q3659962
cites work: Endliche Gruppen I
cites work: Q4377749
cites work: Eigenvalues of a Laplacian and commutative Lie subalgebras
cites work: Sur les groupes nilpotents et les anneaux de Lie
cites work: Normal \(p\)-complements for finite groups
cites work: A replacement theorem for \(p\)-groups and a conjecture
DOI: 10.1016/J.JALGEBRA.2002.11.003
Recommended article: Limits of Abelian subgroups of finite \(p\)-groups
Recommended article: Abelian subgroups of small index in finite p-groups
Recommended article: Auflösbare Gruppen mit endlichem Exponenten, deren Untergruppen alle subnormal sind. I. (Soluble groups of finite exponent all subgroups of which are subnormal. I)
Recommended article: Finite groups with certain normalizers of Sylow subgroups
Recommended article: Non-vanishing elements in finite groups.
Recommended article: On normalizers of Sylow subgroups in finite groups
Recommended article: Q5866984
Recommended article: Q5130703
Recommended article: Q3544720
Recommended article: A note on pronormal \(p\)-subgroups of finite groups
description: scientific article; zbMATH DE number 2057667