Torsion in Groups of Integral Triangles
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Abstract: Let 0＜γ＜π be a fixed pythagorean angle. We study the abelian group Hr of primitive integral triangles (a,b,c) for which the angle opposite side c is γ. Addition in Hr is defined by adding the angles β opposite side b and modding out by π-γ. The only Hr for which the structure is known is Hπ/2, which is free abelian. We prove that for generalγ, Hr has an element of order two iff 2(1- cosγ) is a rational square, and it has elements of order three iff the cubic (2cosγ)x3-3x2+1=0 has a rational solution 0＜x＜1. This shows that the set of values ofγ for which Hr has two-torsion is dense in [0, π], and similarly for three-torsion. We also show that there is at most one copy of either Z2 or Z3 in Hr. Finally, we give some examples of higher order torsion elements in Hr.
Keywords:
Abelian Groups; Cubic Equations; Examples; Free Abelian; Geometric Constructions; Group Theory; Integral Triangles; Law of Cosines; Primitive; Pythagorean Angles; Pythagorean Triangles; Pythagorean Triples; Rational Squares,
Three-Torsion; Torsion; Torsion-Free; Two-Torsion; Triangle Geometry
Cite this paper:
W. Murray, "Torsion in Groups of Integral Triangles," Advances in Pure Mathematics, Vol. 3 No. 1, 2013, pp. 116-120. doi: 10.4236/apm.2013.31015.
References
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