Finding smallest possible group of matrices containing a given matrix I am trying to understand the process for solving group theory questions.
Let $a=\begin{bmatrix}
1&1\\0&1
\end{bmatrix}$ and $b=\begin{bmatrix}i&0\\0&-i\end{bmatrix}$ - 2 x 2 matrices with complex entries.


*

*Describe the smallest group of 2 x 2 complex matrices containing b.

*Describe the smallest group of 2 x 2 complex matrices containing a.


My answer:
Say for question 1 - To find the smallest group I should take $b$ and multiply it by itself $n$ times until I get the identity matrix. And then the identity matrix and the $b, b^2,..., b^n$ other matrices will be the smallest group of 2x2 matrices possible? Is that the correct way to do it?
What about question 1...multiplying $a$ by itself multiple times is not going to bring me back to the identity? So what is the procedure required for that question?
 A: For $b$, you are correct.
For $a$, consider the following two facts (which you can easily prove):


*

*For all integers $n,m$, we have$$\begin{pmatrix}1&n\\0&1\end{pmatrix}\begin{pmatrix}1&m\\0&1\end{pmatrix}=\begin{pmatrix}1&n+m\\0&1\end{pmatrix}.$$

*For a given integer $n$, we have $$\begin{pmatrix}1&n\\0&1\end{pmatrix}^{-1}=\begin{pmatrix}1&-n\\0&1\end{pmatrix}.$$


From this, can you describe the subgroup generated by $a$?

As noted below in the comments, we have an isomorphism
$$\langle\begin{pmatrix}1&1\\0&1\end{pmatrix}\rangle\cong\mathbb Z,$$
given by sending your generator $a$ to the element $1\in\mathbb Z$. But to get a description of $\langle a\rangle$ as a subgroup of the $2\times 2$ complex matrices, we noted that the above facts imply that
$$\langle a\rangle=\left\{\begin{pmatrix}1&n\\0&1\end{pmatrix}:n\in\mathbb Z\right\}.$$

Now, to make sure you understand all of this, convince yourself that given a complex number $\mu\in\mathbb C$, we have the following description:
$$\langle \begin{pmatrix}1&\mu\\0&1\end{pmatrix}\rangle=\left\{\begin{pmatrix}1&n\mu\\0&1\end{pmatrix}:n\in\mathbb Z\right\}.$$
