On the squares of sets of linear groups
A factorization of elements in PSL(2, F), where F = Q, R
Let G be a group and Kₙ = {g ∈ G: o(g) = n}. It is prowed: (i) if F = ℝ, n ≥ 4, then PSL(2,F) = Kₙ²; (ii) if F = ℚ,ℝ, n = ∞, then PSL(2,F) = Kₙ²; (iii) if F = ℝ, then PSL(2,F) = K₃³; (iv) if F = ℚ,ℝ, then PSL(2,F) = K₂³ ∪ E, E ∉ K₂³, where E denotes the unit matrix; (v) if F = ℚ, then PSL(2,F) ≠ K₃³.
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