Let $G$ be a group and let $Z(G)=\{h \in G: g h=h g$ for all $g \in G\}$. Show that $Z(G)$ is a normal subgroup of $G .$

Let $H$ be the set of all $3 \times 3$ real matrices of the form

$$\left(\begin{array}{lll} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{array}\right)$$

with $x, y, z \in \mathbb{R}$. Show that $H$ is a subgroup of the group of invertible real matrices under multiplication.

Find $Z(H)$ and show that $H / Z(H)$ is isomorphic to $\mathbb{R}^{2}$ with vector addition.