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Problem (Practice With Homomorphisms from $\mathbb{Z}_n$ to $\mathbb{Z}_d$)

Let $G=\mathbb{Z}_{12}$ and $H=\mathbb{Z}_3$. Consider the map $f:G\to H$ defined by $f(x)=x\mod 3$.

  1. Show that $f$ is a homomorphism. What is the kernel $K$ of $f$.
  2. Draw Cayley graphs of $G$ and $H$ using the generating set $S=\{1\}$ in both cases, and then draw the identification graph of $G$ using right (or left) cosets of $K$.
  3. How many elements are in $G$, $H$, and $K$? What relationship is there between these?
  4. Repeat parts 1-3 if you use $H=\mathbb{Z}_2$ with $f(x)=x\mod 2$.
  5. Repeat parts 1-3 if you use $H=\mathbb{Z}_4$ with $f(x)=x\mod 4$.
  6. Repeat parts 1-3 if you use $H=\mathbb{Z}_6$ with $f(x)=x\mod 6$.
  7. Pick a positive integer $n$ and a divisor $d$ of $n$. Then repeat 1-3 if $G=\mathbb{Z}_n$, $H=\mathbb{Z}_d$, and $f$ is defined by $f(x)=x\mod d$.

You're welcome to stop when you see a general pattern, provided you describe the general pattern if $G=\mathbb{Z}_{n}$ and $H=\mathbb{Z}_d$.

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