Question

Prove that$\left\{\left(\begin{array}{cc}1& n\\ 0& 1\end{array}\right)|\mathrm{n}\in \mathrm{Z}\right\}$is a cyclic subgroup of GL (2,$\square$)

Short answer

It has been proved that$\left\{\left(\begin{array}{cc}1& n\\ 0& 1\end{array}\right)|\mathrm{n}\in \mathrm{Z}\right\}$is a cyclic subgroup of GL (2,$\square$) .

Step-by-step solution

Find a generator of the group

It is clear that $\left\{\left(\begin{array}{cc}1& n\\ 0& 1\end{array}\right)|\mathrm{n}\in \mathrm{Z}\right\}$is a subgroup of GL (2,$\square$)

To prove that this subgroup is cyclic, we shall find a generator of this subgroup.

Let$g=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$

Using induction, we can see that$g^n=\left(\begin{array}{cc}1& n\\ 0& 1\end{array}\right)$ for every integer n.

Conclusion

We see that g is the generator of the given subgroup.

Hence, it is a cyclic subgroup.