Chapter 3.5, Problem 9E

Find an isomorphism $\varnothing$ from the multiplicative group $G$ of nonzero complex number to the multiplicative group

$H=\left\{\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]|a,b\in ℝand{a}^2+b^2\ne 0\right\}$

and prove that $\varnothing (xy)=\varnothing (x)\varnothing (y)$.

Sec. $3.3,\#15a\gg$

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

a. $H=\left\{\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]|a^2+b^2\ne 0\right\}$