Ambrosiewicz, Jan On the squares of sets of linear groups. (English) Zbl 0591.20048 Rend. Sem. Mat. Univ. Padova 75, 253-256 (1986). 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). Reviewer: Yu.I.Merzlyakov Cited in 1 ReviewCited in 1 Document MSC: 20G15 Linear algebraic groups over arbitrary fields 20E22 Extensions, wreath products, and other compositions of groups 20F05 Generators, relations, and presentations of groups Keywords:\(GL_ n\); \(SL_ n\); \(PSL_ n\); property W PDFBibTeX XMLCite \textit{J. Ambrosiewicz}, Rend. Semin. Mat. Univ. Padova 75, 253--256 (1986; Zbl 0591.20048) Full Text: Numdam EuDML References: [1] J.L. Brenner - L. Carlitz , Covering theorems for finite nonabelian simple groups, III , Rend. Sem. Mat. Uni. Padova, 55 ( 1976 ). Numdam | MR 457549 | Zbl 0352.20003 · Zbl 0352.20003 [2] E. Ambrosiewicz , On groups having the property W, Demonstratio Mathematica , 10 ( 1977 ), pag. 2 . MR 460475 | Zbl 0371.20031 · Zbl 0371.20031 [3] T.W. Hungeerford , Algebra , Springer-Verlag , New York , Heidelberg , Berlin , 1974 . [4] B. Huppert , Endliche Gruppen I , Springer-Verlag , Berlin , Heidelberg , New York , 1967 . MR 224703 | Zbl 0217.07201 · Zbl 0217.07201 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.