I have two questions here, I believe I did them correctly I just want to make sure.
- Find all homomorphisms from $Z_4$ to $Z_6$
- Let $θ$ : ($Z_9$ ,·) → ($Z11)$,·) be a non-trivial homomorphism. Find all elements of $\ker(θ)$ and $\operatorname{Im}(θ)$
For #1 let $Φ$ be from $Z_4$ to $Z_6$. Since $\gcd(4,6)=2$ then there are 2 homomorphisms. We see $4*Φ(1)=(1+1+1+1)=Φ(0)$ so the order of $Φ(1)$ has to divide $4$. Since $Φ(1)$ exists in $\Bbb Z_6$, the order of $Φ$(1) has to divide $6$ as well. So the order of $Φ(1)$ has to be either $1$ or $2$. The only element in $\Bbb Z_6$ of order $1$ is $0$ and there is only one element of order $2$ in $\Bbb Z_6$ and that is $3$. Thus $Φ(1)=0$ or $3$.
For #2 since $θ$ is a homomorphism then by the 1st iso thm $\Bbb Z_9/\ker(θ)$ is isomorphic to $\operatorname{Im}(θ)$. The $|\operatorname{Im}(θ)|$ divides$\gcd(9,11)$ which is $1$. Hence $|\operatorname{Im}(θ)|=1$. By Lagrange's Thm $|\Bbb Z_9/\ker(θ)|=|\operatorname{Im}(θ)|$ which we see is $9/|\ker(θ)|=1$ hence $|\ker(θ)|=9$ and $\ker(θ)=\Bbb Z_9$.
If there is any mistakes or anything that should be added?