# Finite group homomorphism

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.

• $H$ does not have to be finite (e.g. consider $H=G\times\mathbb{Z}$) but the image of $f$ has to be finite. – freakish Jul 29 '19 at 8:28

No. Take, for instance, $$G=(\{-1,1\},\times)$$, let $$H=(\mathbb{Q}\setminus\{0\},\times)$$, and let $$f$$ be the inclusion of $$G$$ into $$H$$.