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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 a

If anyone can clear me up on this that would be much appreciated.

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 a

^{3}and a^{2}a different representatives for the same element? I think I'm confused since a^{2}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 a^{3}and 2 for a^{2}a.If anyone can clear me up on this that would be much appreciated.

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