Let H be the subgroup of $GL(3, \mathbb{Z}_3)$ consisting of all matrices of the form $\left[ \begin{array}{ccc} 1 & 0 & 0 \\ a & 1 & 0 \\ b & c & 1 \end{array} \right]$, where $a,b,c \in \mathbb{Z}_3$. I have to prove that Z(H) is isomorphic to $\mathbb{Z}_3$ and that $H/Z(H)$ is isomorphic to $\mathbb{Z}_3 \times \mathbb{Z}_3$.
I'm really not sure how to begin with this. I started by taking two arbitrary matrices h and k from H and doing hk = kh to see what a matrix in Z(H) would have to look like, but I didn't really get anywhere with that. My initial instinct would be to just define a mapping from Z(H) to $\mathbb{Z}_3$, but I'm not sure how to do that, since I can't figure out what's in Z(H). Is there a better way to do this?
2. You may start with that, i.e. equating the products hk and kh, the only interesting equation you should get is $b+ca'+b' = b'+c'a+b$ (dashed symbols are components of k) then h is in the center iff $ca'=c'a$ holds for any $a',c'\in\mathbb{Z}_3$, so you deduce that $c=a=0$. You can see now why $Z(H)\simeq \mathbb{Z}_3$.
For the second question, it's easy to check that the order of any element of $H$ divides 3. So the order of any element of $H/Z(H)$ divides 3 i.e. it's not cyclic. Conclude.