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
On homomorphisms of a group $G$ to a cyclic group
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group-theory
2 Answers
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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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1@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 $a$.