Question

give an example of a non-trivial homorphiem or explain why none exists.  φ : Z₁₂ → Z₄

Answer 1

$\phi : Z_{12}\rightarrow Z_{4}\\ \\ \\ 1 \rightarrow \phi(1)\\ a \rightarrow \phi(a)=a \phi(1)$

It is sufficient to find the value of $\phi(1)$.

Now $o(\phi(1))$ divides 4 and 12. [ o$(\phi(1))|o(1)$ and o(1)=12]

Therefore $o(\phi(1))=1\hspace{0.5em} \text{or} \hspace{0.5em} 2 \hspace{0.5em}\text{or} \hspace{0.5em} 4$

Now when $o(\phi(1))=1 \Rightarrow \phi(1)=1$ ,

when $o(\phi(1))=2 \Rightarrow \phi(1)=2$ ,

when $o(\phi(1))=4 \Rightarrow \phi(1)=3$

Therefore the possible homomorphisms are $a \rightarrow a$ , $a\rightarrow 2a,a\rightarrow 3a$ .

So the nontrivial homomorphisms are $a \rightarrow 2a$ and $a\rightarrow 3a$