H

1. Definition
$H$ is the set of matrices of the form $\begin{pmatrix} 1 & a & b \\0 & 1 & c \\0 & 0 &1 \end{pmatrix}$ with real entries under matrix multiplication such that

(1)
\begin{align} H = \{\begin{pmatrix} 1 & a & b \\0 & 1 & c \\0 & 0 &1 \end{pmatrix}\right|a,b,c \in\mathbb{R}\} \end{align}

2. Show $H$ is a group
1) Since matrix multiplication is associative, it's associative

2) There is identity matrix $\begin{pmatrix} 1 & 0 & 0 \\0 & 1 & 0\\0 & 0 &1 \end{pmatrix}\in H$ such that $AI=A$ for all $A\in H$

3) There is inverse matrix $A^-1=\begin{pmatrix} 1 & -a & ac-b \\0 & 1 & -c\\0 & 0 &1 \end{pmatrix}\in H$ such that $AA^-1=I$

3. Find all subgroup of $H$
$\{I\}$ $\{I,A^-1,A\}$ $\{I,A^-1,A,AA,AAA,AAAA,....\}$