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### Problem 6

For the following, give an example of a nontrivial homomorphism $\phi$ for the given groups, if an example exists. If no such homomorphism exists, explain why that is so.

(a) (+) $\phi:\mathbb{Z}_{12}\rightarrow \mathbb{Z}_5$

(b) $\phi : \mathbb{Z} \rightarrow S_{3}$

(c) $\phi : \mathbb{Z_{12}} \rightarrow \mathbb{Z_{4}}$

**Solution**

(a) No nontrivial homomorphism exists.

Suppose there exists a homomorphism $\phi:\mathbb{Z}_{12}\rightarrow \mathbb{Z}_5$.

$\forall a\in\mathbb{Z}_{12}$,$|\phi(a)|$ should divide $|a|$ by the homomorphism properties.

$|a|$ could be 1,2,3,4,6,12.

$|\phi(a)|$ could be 1,5.

The only one can be true is a trivial homomorphism.

So there is no nontrivial homomorphism.

(b) A non trivial homomorphism exists, define $\phi (z)=(1,2)^{z}$ where $z \in \mathbb{Z}$. Now, all even integers map to the identity and all odd integers map to $(1,2)$.

(c) A nontrivial homomorphism exists. Define $\phi$ to be the remainder when elements in $\mathbb{Z_{12}}$ are divided by $4$.

For example:

$\bar{6} \rightarrow \bar{2}$

Make sure you can show this is indeed a homomorphism.