I have the following question :
Find all non-trivial homomorphism $\mathbb{Z_{14}}\rightarrow \mathbb{Z_{21}}$ I wonder how to solve such problems when the groups are pretty big to check the definition of each possibility.
Any ideas?
Thank you.
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Have you thought about the image of such a homomorphism? Probably you have some theorems about that. – pjs36 Jan 30 '17 at 13:57
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1If $C$ is a cyclic group generated by $\sigma$ and $G$ is any other group then homomorphisms $C\to G$ correspond to places to send $\sigma$ to... – anon Jan 30 '17 at 13:57
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1Also an answer here. – Dietrich Burde Jan 30 '17 at 14:01
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$\phi : \mathbb{Z}_n \to G$ is an homomorphism iff $ord(\phi(1)) \ \mid \ n$, and it is non-trivial iff $ord(\phi(1)) > 1$. – reuns Jan 30 '17 at 15:00