1.
First, what are the possible kernels of such a homomorphism? Remember, all subgroups of a cyclic group are cyclic, so you can write subgroups of \(\Z_{12}\) as \(\langle 4 \rangle = \{0, 4, 8\}\text{,}\) for example.
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Worksheet Homomorphic Images
To get a feel for what the homomorphic image of a group can be, consider the following question:
2.
Find a homomorphism \(f:\Z_{12} \to \Z_{20}\) that has \(\langle 4 \rangle\) as its kernel. How large will the range (i.e. image) of \(f\) be?
3.
What is the quotient group \(\Z_{12}/\langle 4 \rangle\) and how does this relate to the image of the homomorphism?
4.
Which other subgroups of \(\Z_{12}\) could actually be kernels of some homomorphism? Find a subgroup that does NOT work. What goes wrong?
