# Isomorphism, Cyclic Groups, and Generators

Prove that any isomorphism between two cyclic groups always maps every generator to a generator.

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Please don't formulate your questions in the imperative, it is considered impolite. You should also say what you have tried already and where your difficulties lie. If you have no idea at all, then you would be better off asking for help on the fundamental concepts and not on specific questions. And finally, when posting homework questions, please use the (homework) tag. –  Alex B. Nov 30 '10 at 2:07
Thank you Alex. –  user4179 Dec 1 '10 at 2:35

Step 1: Show that any homomorphism from a cyclic group is determined by its image on a generator of the group.

Step 2: Show that the image of the homomorphism is generated by the image of that generator.

Step 3: What can you say about the kernel of such a homomorphism? What powers of the generator does it send to the identity?

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Suppose you have two cyclic groups $G$ and $G'$ and some isomorphism $\phi\colon G\to G'$. Since $\phi$ is surjective, for any $g'\in G'$ there exists some $g\in G$ such that

$$g'=\phi(g).$$

Try expressing $g$ and/or $g'$ in terms of the generators, and see what happens from there. The fact that $\phi$ is a homomorphism is particularly useful here.

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