On the Magnus Function Roger C. Alperin ¤ 1 Introduction There are uncountably many integer functions M : Z ¡! Z having the properties M(0) = 0;M(M(x)) = x;M(x) = M(M(x ¡ 1) ¡ 1) ¡ 1: Using this B. Neumann [cf. M2] showed the existence of uncountably many maximal non-parabolic subgroups of the modular group PSL2(Z).