Excuse me if I get the English names wrong here, I hope my question is clear. I would be happy if someone could explain this problem to me ... I'm having trouble with some aspects of homomorphisms. The question is: Is there a homomorphism Phi: Z24 --> Z18 which fullfills Phi(1)=16? The book just gives the answer as "no", but I don't really know how to figure this out in any direct way. We had an earlier similar problem where we were given a similar homomorphism that we were told was a homomorphism: Z18-->Z12 where phi(1)=8, though no more details than that. We were only asked to find im phi, and ker phi, which I understood how to do. But it seems to me to be about the same for the above problema and would give similar results for ther image and kernel of phi, though obviously all phi will fail some test of being a homomorphism. I would be most happy if someone could explain this to me. Thanks.