What does it mean to say that $H$ is a normal subgroup of the group $G$ ? For a normal subgroup $H$ of $G$ define the quotient group $G / H$. [You do not need to verify that $G / H$ is a group.]

State the Isomorphism Theorem.

Let

$$G=\left\{\left(\begin{array}{ll} a & b \\ 0 & d \end{array}\right) \mid a, b, d \in \mathbb{R}, a d \neq 0\right\}$$

be the group of $2 \times 2$ invertible upper-triangular real matrices. By considering a suitable homomorphism, show that the subset

$$H=\left\{\left(\begin{array}{ll} 1 & b \\ 0 & 1 \end{array}\right) \mid b \in \mathbb{R}\right\}$$

of $G$ is a normal subgroup of $G$ and identify the quotient $G / H$.