# On homomorphisms of a group $G$ to a cyclic group

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

-

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 $G$?

-
@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 $a$.

-