point on the unit circle, where a/c and b/c are reduced fractions, then (c/a, b/a) is a rational point on the unit hyperbola, since satisfying the equation for the unit hyperbola. The group operation here is and the group identity is the same point (1, 0) as above. In this group there is a close connection with the hyperbolic cosine and hyperbolic sine, which parallels the connection with cosine and sine in the unit circle group above. Copies inside a larger group There are isomorphic copies of both groups, as subgroups (and as geometric objects) of the group of the rational points on the abelian variety in four-dimensional space given by the equation Note that this variety is the set of points with Minkowski metric relative to the origin equal to 0. The identity in this larger group is (1, 0, 1, 0), and the group operation is For the group on the unit circle, the appropriate subgroup is the subgroup of points of the form (w, x, 1, 0), with and its identity element is (1, 0, 1, 0). The unit hyperbola group corresponds to points of form (1, 0, y, z), with and the identity is again (1, 0, 1, 0). (Of course, since they are subgroups of the larger group, they both must have the same identity element.) See also Circle group References The Group of Rational Points on the Unit Circle, Lin Tan, Mathematics Magazine Vol. 69, No. 3 (June, 1996), pp. 163–171 The Group of Primitive Pythagorean Triangles, Ernest J. Eckert, Mathematics Magazine Vol 57 No. 1 (January, 1984), pp 22–26 ’’Rational Points on Elliptic Curves’’ Joseph Silverman Abelian group theory