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On the squares of sets of linear groups. (English) Zbl 0591.20048

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

MSC:

20G15 Linear algebraic groups over arbitrary fields
20E22 Extensions, wreath products, and other compositions of groups
20F05 Generators, relations, and presentations of groups
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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
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