Prove $H$ is a normal subgroup of $K$ $G$ is a set.
$$G=\left\{\begin{vmatrix}1 & 0 \\ 0  & 1\end{vmatrix}, \begin{vmatrix}0 & 1 \\-1 & 0\end{vmatrix}, \begin{vmatrix}-1 & 0 \\ 0  & -1\end{vmatrix}, \begin{vmatrix}0 & -1 \\ 1  & 0\end{vmatrix}, \begin{vmatrix}i & 0 \\ 0  & i\end{vmatrix}, \begin{vmatrix}0 & i \\ -i  & 0\end{vmatrix}, \begin{vmatrix}-i & 0 \\ 0  & i\end{vmatrix}, \begin{vmatrix}0 & -i \\ i  & 0\end{vmatrix}\right\}$$ and $(G, \times)$ is a group.
If
$$H=\left\{\begin{vmatrix}1 & 0 \\ 0  & 1\end{vmatrix}, \begin{vmatrix}-1 & 0 \\ 0  & -1\end{vmatrix}\right\}$$ and
$$K=\left\{\begin{vmatrix}1 & 0 \\ 0  & 1\end{vmatrix}, \begin{vmatrix}-1 & 0 \\ 0  & -1\end{vmatrix}, \begin{vmatrix}i & 0 \\ 0  & i\end{vmatrix}, \begin{vmatrix}-i & 0 \\ 0  & -i\end{vmatrix}\right\},$$
then $H$ and $K$ is a subgroup of $G$, $H$ is a normal subgroup of $K$, and $K$ is a normal subgroup of $G$.
My questions are :
a. How do I prove that $H$ is a normal subgroup of $K$?
I knew that to prove if $H$ is a normal subgroup I have to prove that each left coset of $H$ in K is a right coset of $H$ in $K.$ But I don't know how apply that to the matrices.
b. What is every element of factor group $G/H?$

a. $H=\{I,-I\}$, so, for all $k\in K$, $kH=\{kI,-kI\}=\{k,-k\}$,
and also $Hk=\{Ik,-Ik\}=\{k,-k\}$,
so indeed each left coset of $H$ in $K$ is a right coset of $H$ in $K$, so $H\lhd K$.
b.  One element of $G/H$, for example,
is  $\left\{I\begin{bmatrix}0 & i \\ -i  & 0\end{bmatrix},   -I\begin{bmatrix}0 & i \\ -i  & 0\end{bmatrix}\right\}$ =$ \left\{\begin{bmatrix}0 & i \\ -i  & 0\end{bmatrix},   \begin{bmatrix}0 & -i \\ i  & 0\end{bmatrix}\right\}.$
Now can you find the others?
