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Let $G=\mathbb{Z}_{8}\times \mathbb{Z}_{12}\times \mathbb{Z}_{30}$, where $\mathbb{Z}_{n}$ denotes the cyclic group of order $n$. Does $G$ admit a homomorphism onto $\mathbb{Z}_{45}$? What about $\mathbb{Z}_{120}$? Thank you

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HINTS:

(1) $\Bbb Z_{45}$ has an element of order $9$; does $G$?

(2) What is the order of $\langle 1,1,1\rangle$ in $G$?

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    @Babak: That’s not necessarily true unless $\varphi$ is actually an isomorphism. After all, the trivial group is a homomorphic image of **every** group! What we do know is that $|\varphi(g)|$ divides $|g|$.2012-12-30
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Another hint: Prove that if $\phi$ is an homomorphism then the order of $\phi(a)$ divides the order of $a$.