So letLet $\displaystyle G = \{ z \in \mathbb{C} |z^n = 1 \text{ for some n } \in \mathbb{Z} ^+ \}$. Prove that for any fixed integer k > 1 the map from G to itself defined by $\displaystyle z \mapsto z^k$ is a surjective homomorphism but is not injective.

$\displaystyle \phi _k : G \to G : g \mapsto g^k$.

Proving the homomorphism is trivial. It's the surjection part that's getting to me. (I haven't tackled the injection yet. Please leave that out of the replies for now.) What I am looking at is the following...I want to show that there is at least one pair of x, y in G such that $\displaystyle x^k = y^k$ is true when x is not equal to y. My other question about this is do we need to find an x, y for each k? For example, $\displaystyle (-1)^k = (1)^k$ is only true for even k.

I'm just not wrapping this one around in my head today.

Thanks!

-Dan