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Let G be a cyclic group, and H be any group. (i) Prove that any homomorphism...

Let G be a cyclic group, and H be any group. (i) Prove that any homomorphism ϕ : G → H is uniquely determined by where it maps a generator of G. In other words, if G = <x> and h ∈ H, then there is at most one homomorphism ϕ : G → H such that ϕ(x) = h. (ii) Why is there ‘at most one’? Give an example where no such homomorphism can exist.

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