Chapter 3.5, Problem 8E

Find an isomorphism from the group $G=\{1,i,-1,-i\}$ in Example $5$ of this section to the multiplicative group

$H=\left\{\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}i& 0\\ 0& -i\end{array}\right],\left[\begin{array}{cc}-i& 0\\ 0& i\end{array}\right],\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\right\}$.

Sec. $3.3,\#16b\gg$

Prove that each of the following sets $H$ is a subgroup of $GL(2,ℂ)$, the general linear group of order $2$ over $ℂ$.

$H=\left\{\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\right\}$

$H=\left\{\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}i& 0\\ 0& -i\end{array}\right],\left[\begin{array}{cc}-i& 0\\ 0& i\end{array}\right],\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\right\}$