Large Abelian subgroups of finite \(p\)-groups
type: Item
instance of: scholarly article
title: Large Abelian subgroups of finite \(p\)-groups
zbMATH Open document ID: 0917.20016
author: George Glauberman
published in: Journal of Algebra
publication date: 1999-08-02T00:00:00Z
review text: Let \(p\) be a prime and let \(G\) be a finite \(p\)-group. Let \(d(G)\) be the maximum of \(| A|\) as \(A\) ranges over the abelian subgroups of \(G\), and \({\mathcal A}(G)\) be the set of all abelian subgroups \(A\) for which \(| A|=d(G)\). Let \(B\) be an abelian subgroup of \(G\) normalized by \(A\) and \(B\) does not normalize \(A\). It is known that there exists \(A^*\in{\mathcal A}(G)\) such that \(A^*\leq AB\), \(A\cap B<A^*\cap B\) and \(A^*\) normalizes \(A\). The author extends this result by allowing \(B\) to have nilpotence class 2 instead of necessarily being abelian when \(p\) is odd. In particular it is proved, that if \(G\) is a finite \(p\)-group, \(p\) is an odd prime, \(A\in{\mathcal A}(G)\), \(B\) is a subgroup of \(G\) that does not normalize \(A\), \(B\) has nilpotence class at most 2 and is normalized by \(A\) or, if \(p\geq 5\), \(A\triangleleft\langle A^B\rangle\), and \([A,u;3]=1\) for every \(u\in B\), then there exists \(A^*\in{\mathcal A}(G)\) such that \(A^*\leq\langle A^B\rangle\) and \(A^*\) normalizes \(A\) and if \(A\) normalizes \(B\) then \(| A^*\cap B|\geq| A\cap B|\).
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 1089125
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: Abelian subgroups
reviewed by: N. M. Kurnosenko
MaRDI profile type: Publication
full work available at URL: jabr.1997.7090
OpenAlex ID: W2069189626
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DOI: 10.1006/JABR.1997.7090
description: scientific article; zbMATH DE number 1089125