Geodesic
A factorization of elements in PSL(2, F), where F = Q, R
Discussiones Mathematicae. General Algebra and Applications, Tome 20 (2000) no. 2, pp. 159-167.

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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.
Keywords: factorization of linear groups, linear groups, matrix representations of groups, sets of elements of the same order in groups 
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Ambrosiewicz, Jan. A factorization of elements in PSL(2, F), where F = Q, R. Discussiones Mathematicae. General Algebra and Applications, Tome 20 (2000) no. 2, pp. 159-167. http://geodesic.mathdoc.fr/item/DMGAA_2000_20_2_a0/

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