 
Summary: 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 nonparabolic subgroups of the modular group PSL2(Z).
In his article, [M1], Magnus inquires about the existence of functions
M : Z[1/2]  Z[1/2], on the ring of rational numbers with denominator a
power of 2, the dyadic fractions Z[1/2], which have the `Neumann' properties
displayed above and the additional properties,
M(2) = 2, M(x/2) = 2M(x).
Magnus wonders if there are only countably many maximal nonparabolic
subgroups of the group PSL2(Z[1/2]).
If such a Magnus function exists, it is determined by these displayed con
ditions and the values at the odd integers. Assuming existence of a Magnus
function, M, it follows that M(2) = 1
2
M(1) so that M(1) = 4, and also
