The complex numbers {x + iy} (x,y, real) lie on the unit circle when x^2 + y^2 = 1. These form the circle group under multiplication. If in addition x and y are rational numbers, this is a countably infinite subgroup of the circle group. What is its structure as a group? That problem was solved in this paper: <https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1997/0025570x.di021195.02p0087x.pdf>, which shows (Theorem 1) that the group is isomorphic to the direct sum of the group Z/4Z and countably many infinite cyclic groups. Likewise, the set of quaternions {x + iy + zj + wk} (x,y,z,w real) with x^2 + y^2 + z^2 + w^2 = 1 forms a group under multiplication: it is the versatile group S^3 of unit quaternions that is isomorphic also to the groups Spin(3) and SU(2).* Once again, if we restrict x, y, z, w to be rational numbers, this becomes a countable subgroup of S^3. Question: What is the structure of this group of rational unit quaternions? —Dan ————— * Spin(3) is the double cover of the rotation group SO(3) or R^3, and SU(2) is the group of 2x2 complex unitary matrices** having determinant = 1. ** The conjugate transpose is the inverse matrix.