  Let f be a function from the set of rational numbers into itself. We call f a
global power map if f(n) = n^k for some integer exponent k. We call f a local
power map at the prime number p if f induces a well-defined group homomorphism
on the multiplicative group of integers modulo p. We conjecture that if f is a
local power map at an infinite number of primes p, then f must be a global
power map. Our main theorem implies that if f is a local power map at every
prime p in a set with positive upper density relative to the set of all primes,
then f must be a global power map. In particular, this represents progress
towards a conjecture of Fabrykowski and Subbarao.
