Chapter 3.5, Problem 11E

The following set of matrices

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

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

forms a group $H$ with respect to matrix multiplication. Find an isomorphism from $H$ to the quaternion group.

Sec. $3.1,\#34\gg$

Let $G$ be the set of eight elements $G=\left\{1,i,j,k,-1,-i,-j,-k\right\}$ with identity element $1$ and noncommutative multiplication given by

${\left(-1\right)}^2=1,$

$i^2=j^2=k^2=-1,$

$ij=-ji=k$

$jk=-kj=i,$

$ki=-ik=j,$

$-x=\left(-1\right)x=x\left(-1\right)$ for all $x$ in $G$

(The circular order of multiplication is indicated by the diagram in Figure $3.8$.) Given that $G$ is a group of order $8$, write out the multiplication table for $G$. This group is known as the quaternion group.