On the squares of sets of linear groups
type: Item
instance of: scholarly article
title: On the squares of sets of linear groups
zbMATH Open document ID: 0591.20048
author: Jan Ambrosiewicz
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1986-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108025
review text: For a given group G and a natural number w, let \(K_ w\) denote the set of elements of order w in G. By definition, the group G has the property W if, for every w, the product \(K_ wK_ w\) is a subgroup of G. It is proved in this article that the groups \(GL_ n(k)\), \(SL_ n(k)\) and \(PSL_ n(k)\) with \(n\geq 3\) over any field k have not the property W (Theorem 1). If char\(k=2\) and \(| k| >2\) then \(SL_ 2(k)=K_ 2K_ 2\) (Theorem 2); if char \(k\neq 2\) then \(K_ 2K_ 2\) is not a subgroup in \(GL_ 2(k)\) (Theorem 3). If char \(k\neq 2\) and the element -1 is a square in the field k then \(PSL_ 2(k)=K_ 2K_ 2\); if char \(k\neq 2\), \(| k| \neq 3\) and the element -1 is not a square in k then \(K_ 2K_ 2\) is not a subgroup in \(PSL_ 2(k)\) (Theorem 5).
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 3948599
zbMATH Keywords: \(GL_ n\)
zbMATH Keywords: \(SL_ n\)
zbMATH Keywords: \(PSL_ n\)
zbMATH Keywords: property W
reviewed by: Ju. I. Merzljakov
MaRDI profile type: Publication
cites work: Covering theorems for finite nonabelian finite simple groups. III. Solutions of the equation \(\alpha x^ 2 + \beta t^ 2 + y t^{-2} =\) a in a finite field
cites work: On Groups Having The Property W
cites work: Endliche Gruppen I
description: scientific article; zbMATH DE number 3948599