Please Login to access more options.

Problem (Problem.Practice With Identification Graphs Of $\mathbb{Z}$)

Let $G=\mathbb{Z}$ and consider the subgroup $N=3\mathbb{Z}$.

  1. There are infinitely many possible ways to write cosets of $N$, namely we have $0+3\mathbb{Z}$, $1+3\mathbb{Z}$, $2+3\mathbb{Z}$, $3+3\mathbb{Z}$, $4+3\mathbb{Z}$, etc. Show that there are finitely many different cosets.
  2. Draw Cayley graphs of $G$ using the generating set $S=\{1\}$ in both cases, and then draw the identification graph of $G$ using right (or left) cosets of $N$. As there are only finitely many cosets, you should obtain a finite graph for the identification graph.
  3. Can you think of a group $H$ and a homomorphism $f:\mathbb{Z}\to H$ so that the kernel of $f$ is the subgroup $N=3\mathbb{Z}$? Hint: look at the previous problem.
  4. Repeat parts 1-3 if you use $H=\mathbb{Z}_5$ with $f(x)=x\mod 5$.
  5. Repeat parts 1-3 if you use $H=\mathbb{Z}_n$ with $f(x)=x\mod n$.
  6. What does this problem have to do with modular arithmetic?

The following pages link to this page.