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 $G/H?$

as for $H$ in $K$ you could notice that $H$ is of index 2 in K so it is normal.
