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I am trying to figure out all of the homomorphisms between these groups.

First by homomorphisms properties, the identity of $\Bbb Z_{24}$ is mapped to $\Bbb Z_{18}$ (i.e $\phi(0) = 0$).

After that, I got a bit confuse since the different combinations of mapping between the groups do not seems clear to me. I tried to list all of the subgroups of both groups but I do not know how these informations are helpful to me.

Thank you,

Shaun
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Arthur Madore Boisvert
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    Is there a generator of $\Bbb{Z}/24\Bbb{Z}$? A generator has order the same as the group. Suppose $g$ is such a generator. Once you specify $\varphi(g)$, do the properties of homomorphisms force the image under $\varphi$ of every other element of $\Bbb{Z}/24\Bbb{Z}$? There are at most $18$ place you can send $g$ in $\Bbb{Z}/18\Bbb{Z}$. What happens with each of them? – Eric Towers Dec 13 '21 at 20:59
  • yes so in this case, the homorphisms between the groups would be $ \phi(n)=xn$ where $x =0,4,8,12,16,20$ ? – Arthur Madore Boisvert Dec 14 '21 at 00:28

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