Let \(G=\left{z\in\mathbb{C}|z^n=1\text{ for some }n\in\mathbb{Z}^+\right}\). Prove that for any fixed integer \(k > 1\) the map from G to itself defined by \(z \mapsto z^k\) is a surjective homomorphism but not an isomorphism.
I can show that it's a homomorphism, but am a bit stuck after that

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isomorphism requires both injective and surjective homomorphism

I know that but I wouldn't have a clue how to prove this is not injective :)

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