Rationals and the Modular Group Roger C. Alperin Summary: Rationals and the Modular Group Roger C. Alperin The modular group k is the quotient group PSL2(Z) = SL,(Z)/{&I} of SL2(Z), the group of 2 x 2 integer matrices of determinant 1. In [I]we gave an elementary proof that .A' has the structure of a free product of a cyclic group of order 2 generated by the image of A = (:-:,) and a cyclic group of order 3 generated by theimageof B = (i :I.The free product structure provides a description of the non-trivial elements of k as unique strings of A's and B's with the property that there are no two consecutive A's and no three consecutive B7s;we refer to these as reduced strings. We explained this free product structure in terms of the action of the modular group on the irrationals. In this note we describe the action on the rationals; this can be viewed as a way of describing the inverse of the Euclidean algorithm. The group SL2(Z)acts via linear transformations on R2 as column vectors and this gives an action of .A' via linear fractional transformations on the projective line P'(R), the real numbers together with m. We may also view P'(R) as the slopes of non-zero vectors, that is, the equivalence classes of R2 - (:)induced by non-zero scalar multiplication; the equivalence class of the vector e = (i) is , , denoted x, the equivalence class of the vector , q # 0 is the same as that of (':I and corresponds to the real number z For the matrix (::)in Collections: Mathematics