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**1. The problem statement, all variables and given/known data**

Let H be the subgroup of [tex]GL(3, \mathbb{Z}_3)[/tex] consisting of all matrices of the form [tex]\left[ \begin{array}{ccc} 1 & 0 & 0 \\ a & 1 & 0 \\ b & c & 1 \end{array} \right][/tex], where [tex]a,b,c \in \mathbb{Z}_3[/tex]. I have to prove that Z(H) is isomorphic to [tex]\mathbb{Z}_3[/tex] and that [tex]H/Z(H)[/tex] is isomorphic to [tex]\mathbb{Z}_3 \times \mathbb{Z}_3[/tex].

**2. Relevant equations**

**3. The attempt at a solution**

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 [tex]\mathbb{Z}_3[/tex], 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?