the complex plane, and thus are elements of G. Their group product is −33/65 + 56/65i, which corresponds to the Pythagorean triple (33,56,65). The sum of the squares of the numerators 33 and 56 is 1089 + 3136 = 4225, which is the square of the denominator 65. Other ways to describe the group The set of all 2×2 rotation matrices with rational entries coincides with G. This follows from the fact that the circle group is isomorphic to , and the fact that their rational points coincide. Group structure The structure of G is an infinite sum of cyclic groups. Let G2 denote the subgroup of G generated by the point . G2 is a cyclic subgroup of order 4. For a prime p of form 4k + 1, let Gp denote the subgroup of elements with denominator pn where n is a non-negative integer. Gp is an infinite cyclic group, and the point (a2 − b2)/p + (2ab/p)i is a generator of Gp. Furthermore, by factoring the denominators of an element of G, it can be shown that G is a direct sum of G2 and the Gp. That is: Since it is a direct sum rather than direct product, only finitely many of the values in the Gps are non-zero. Example Viewing G as an infinite direct sum, consider the element ({0}; 2, 0, 1, 0, 0, ..., 0, ...) where the first coordinate 0 is in C4 and the other coordinates give the powers of (a2 − b2)/p(r) + i2ab/p(r), where p(r) is the rth prime number of form 4k + 1. Then this corresponds to, in G, the rational point (3/5 + i4/5)2 · (8/17 + i15/17)1 = −416/425 + i87/425. The denominator 425 is the product of the denominator 5 twice, and the denominator 17 once, and as in the previous example, the square of the numerator −416 plus the square of the numerator 87 is equal to the square of the denominator 425. It should also be noted, as a connection to help retain understanding, that the denominator 5 = p(1) is the 1st prime of form 4k + 1, and the denominator 17 = p(3) is the 3rd prime of form 4k + 1. The unit hyperbola's group of rational points There is a close connection between this group on the unit hyperbola and the group discussed above. If is a rational