Torsion in Groups of Integral Triangles

Will Murray

Department of Mathematics and Statistics, California State University, Long Beach, USA.

DOI: 10.4236/apm.2013.31015                

Abstract

Let 0＜γ＜π be a fixed pythagorean angle. We study the abelian group H_r of primitive integral triangles (a,b,c) for which the angle opposite side c is γ. Addition in H_r is defined by adding the angles β opposite side b and modding out by π-γ. The only H_r for which the structure is known is H_π/2, which is free abelian. We prove that for generalγ, H_r 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 H_r 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 H_r. Finally, we give some examples of higher order torsion elements in H_r.

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

Conflicts of Interest

The authors declare no conflicts of interest.

References

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