I am trying to show show that there is no homomorphism from Zp1 to Zp2. if p1 and p2 are different prime numbers. (Zp1 and Zp2 represent cyclic groups with addition mod p1 and p2 respectively). I am not sure how to do this but here are some thoughts; For there to be a homomorphism we require: F(g.h)=F(g)F(h) * where F maps Zp1 to Zp2. From the information in the question - we can say the following (not sure if it is relavent): Since p1 is a prime number, from Lagranges Theorem, there are no (non trivial) subgroups in Zp1, so every pair of elements in this group must satisfy *. And because z1 and z2 are unequal, the homomorphism must be such that more than one element of one group maps to just one element of the other. Any thoughts on where to go from here?