A group of upper-triangular matrices

#group_theory

Let $G$ be the group of upper-triangular real matrices $[\begin{matrix} a & b \\ 0 & d \end{matrix}]$ with $a, d \neq 0$ , under matrix multiplication. Let $S$ be the subset of $G$ defined by $d = 1$ . Show that $S$ is normal and that $G / S ≅ R^{\times}$ , the multiplicative group of nonzero real numbers.

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Let $G$ be the group of upper-triangular real matrices $\begin{bmatrix} a & b \\ 0 & d\end{bmatrix}$ with $a,d\neq 0$, under matrix multiplication. Let $S$ be the subset of $G$ defined by $d=1$. Show that $S$ is normal and that $G/S\cong {\bf R}^{\times}$, the multiplicative group of nonzero real numbers.