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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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up vote 1 down vote accepted


(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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@maryam: Does we know that if $\phi:G\to G'$ a homomorphism, then $|\phi(g)|\big||g|$? – Babak S. Dec 30 '12 at 13:21
@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 $|g|$. – Brian M. Scott Dec 30 '12 at 13:23

Another hint: Prove that if $\phi$ is an homomorphism then the order of $\phi(a)$ divides the order of $a$.

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