On the squares of sets of linear groups
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).
scientific article; zbMATH DE number 3948599