Prove that any isomorphism between two cyclic groups always maps every generator to a generator.
Isomorphism, Cyclic Groups, and Generators
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abstract-algebra
group-theory
2 Answers
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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.