TITLE: Is this group of matrices cyclic?
QUESTION [6 upvotes]: Is the group $H$ consisting of matrices of the form 
$ 
  \left( {\begin{array}{cc}
   1 & n \\       0 & 1 \\      \end{array} } \right)
$ cyclic, where $n \in \mathbb{Z}$? If not, how would you show this?

REPLY [7 votes]: if we define:
$$
B=\left( {\begin{array}{cc}
   0 & 1 \\       0 & 0 \\      \end{array} } \right)
$$
then
$$
A = I+B \\
B^2 = 0
$$
so by the binomial theorem, for $n \ge 0$:
$$
A^n =(I+B)^n = I+nB
$$
note also that:
$$
(I+B)(I-B) = I-B^2=I
$$
so
$$
A^{-1}=I-B \\
A^{-n}=(I-B)^n =I-nB
$$