Proving Isomorphism of Heisenberg Group over Finite Field

[dancavallaro]

Homework Statement

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 $$\mathbb{Z}_3 \times \mathbb{Z}_3$$.

Homework Equations

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 $$\mathbb{Z}_3$$, 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?

 

[rsg]

These groups are known in literature as Heisenberg group over a finite field. It is generally written in the upper triangular matrix form. But that does not make any difference.
So let me define a typical element of this field as $$A=\langle a,b,c\rangle$$ where a, b, c have the same meaning and order as you have written in the definition of the group. Let $$C=\langle x,y,z\rangle\in Z(H)$$. Calculate $$AC$$ and $$CA$$, and equate them...


Edited by Hurkyl: please don't give complete solutions to problems...