Elementary Methods in Number Theory Exercise 1.2.25

Let $G$ be the set of all matrices of the form
\begin{equation}
\begin{pmatrix}
1&a\\
0&1\\
\end{pmatrix}
\end{equation}
with $a\in \mathbf{Z}$ and matrix multiplication as the binary operation.Prove that $G$ is an abelian group isomorphic to $\mathbf{Z}$.

 

Proof:
\begin{equation}
\begin{pmatrix}
1&a_1\\
0&1\\
\end{pmatrix}\begin{pmatrix}
1&a_2\\
0&1\\
\end{pmatrix}=\begin{pmatrix}
1&a_2+a_1\\
0&1\\
\end{pmatrix}
\end{equation}
Done.