Torsion in Groups of Integral Triangles


Let 0γ<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 0x1. 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.

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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.

Conflicts of Interest

The authors declare no conflicts of interest.


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