Let $H$ be the Heisenberg group. Determine the center $Z(H)$ of $H$. Show that the quotient group $H/Z(H)$ is abelian.

Question

Let $H$ be the Heisenberg group. Determine the center $Z(H)$ of $H$. Show that the quotient group $H/Z(H)$ is abelian.

Here

Show that the quotient of the Heisenberg Group with its center is abelian.

also it is given as an exercise.

I am new in this matrix algebra field. I was trying to prove the first part using elementary matrix but it does not work out. May be, I am wrong at some point any help would be appreciated.

Answer 1

Let$$\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix}\in Z(H).\tag1$$Then, if $x,y,z\in\mathbb{R}$, then$$\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix}^{-1}\begin{bmatrix}1&x&z\\0&1&y\\0&0&1\end{bmatrix}\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix}=\begin{bmatrix}1&x&z\\0&1&y\\0&0&1\end{bmatrix}.$$But$$\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix}^{-1}\begin{bmatrix}1&x&z\\0&1&y\\0&0&1\end{bmatrix}\begin{bmatrix}1&a&c\\0&1&b\\0&0&1\end{bmatrix}=\begin{bmatrix}1&x&ay-bx+z\\0&1&y\\0&0&1\end{bmatrix}.$$Therefore, $a=b=0$ if and only if $(1)$ holds. So, now you know $Z(H)$. Can you take it from here?