So this is a pretty dumb question, but I'm just trying to understand homomorphisms of infinite cyclic groups. I understand intuitively why if we define the homomorphism p(a)=b, then this defines a unique homorphism. My question is why is it necessarily well-defined? I think I'm confused because I'm not sure what a 'representation' of an element really means. For example, in the quotient group, we take any element in the equivalence class [a] to be a representative of [a], and we have to show that the homomorphism acts the same on every representative (which it doesn't in some cases). So, are a3 and a2a different representatives for the same element? I think I'm confused since a2 is just notational convention for aa, so I'm not sure if this really constitutes a different 'representative'. What if I defined f(x)="first power that appears in x". Would that be 3 for a3 and 2 for a2a. If anyone can clear me up on this that would be much appreciated.