Q:

Group homomorphism question.

T/F? There exist two groups $G$ and $H$ such that $|G|=12, |H|=8$, and there is a surjective homomorphism $\phi: G \to H$.
Answer says "False, because $|\phi [G]|$ must divide $|G|$."
Can someone explain this? Thank you.

A:

Let $G$ and $H$ be groups with $|G|=12$ and $|H|=8$. Seeking a contradiction, suppose there exists a surjective group homomorphism $\phi:G\rightarrow H$. Then the first isomorphism theorem ensures a group-isomorphism $H\simeq G/\mathrm{Ker}\,\phi$. It follows that $8=12/|\mathrm{Ker}\,\phi|$ so that $|\mathrm{Ker}\,\phi|=3/2$, a contradiction. Hence there exists no surjective group homomorphism $G\rightarrow H$.

