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I have a question about the group homomorphism. Let $G$ is finite group. $f:G\to H$ is homomorphism. Does $H$ have to be a finite set ? May you explain with example?

Thank you.

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  • $\begingroup$ $H$ does not have to be finite (e.g. consider $H=G\times\mathbb{Z}$) but the image of $f$ has to be finite. $\endgroup$ – freakish Jul 29 '19 at 8:28
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No. Take, for instance, $G=(\{-1,1\},\times)$, let $H=(\mathbb{Q}\setminus\{0\},\times)$, and let $f$ be the inclusion of $G$ into $H$.

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