So letLet . Prove that for any fixed integer k > 1 the map from G to itself defined by is a surjective homomorphism but is not injective.
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 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, is only true for even k.
I'm just not wrapping this one around in my head today.