Torsion in Groups of Integral Triangles

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 general γ , H r has an element of order two iff 2(1- cos γ ) is a rational square, and it has elements of order three iff the cubic (2cos γ ) x 3 -3 x 2 +1=0 has a rational solution 0 ＜ x ＜ 1 . This shows that the set of values of γ 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 Z 2 or Z 3 in H r . Finally, we give some examples of higher order torsion elements in H r .