$\mathbf { 4 }$ The group $G$ consists of the 8 complex matrices $\{ \mathbf { I } , \mathbf { J } , \mathbf { K } , \mathbf { L } , - \mathbf { I } , - \mathbf { J } , - \mathbf { K } , - \mathbf { L } \}$ under matrix multiplication, where

$$\mathbf { I } = \left( \begin{array} { l l } 
1 & 0 \\
0 & 1
\end{array} \right) , \quad \mathbf { J } = \left( \begin{array} { r r } 
\mathrm { j } & 0 \\
0 & - \mathrm { j }
\end{array} \right) , \quad \mathbf { K } = \left( \begin{array} { r r } 
0 & 1 \\
- 1 & 0
\end{array} \right) , \quad \mathbf { L } = \left( \begin{array} { c c } 
0 & \mathrm { j } \\
\mathrm { j } & 0
\end{array} \right)$$

(i) Copy and complete the following composition table for $G$.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
 & I & J & K & L & -I & -J & -K & $- \mathbf { L }$ \\
\hline
I & I & J & K & L & -I & -J & -K & -L \\
\hline
J & J & -I & L & -K & -J & I & -L & K \\
\hline
K & K & -L & -I &  &  &  &  &  \\
\hline
L & L & K &  &  &  &  &  &  \\
\hline
-I & -I & -J &  &  &  &  &  &  \\
\hline
-J & -J & I &  &  &  &  &  &  \\
\hline
-K & -K & L &  &  &  &  &  &  \\
\hline
-L & -L & -K &  &  &  &  &  &  \\
\hline
\end{tabular}
\end{center}

(Note that $\mathbf { J K } = \mathbf { L }$ and $\mathbf { K J } = - \mathbf { L }$.)\\
(ii) State the inverse of each element of $G$.\\
(iii) Find the order of each element of $G$.\\
(iv) Explain why, if $G$ has a subgroup of order 4, that subgroup must be cyclic.\\
(v) Find all the proper subgroups of $G$.\\
(vi) Show that $G$ is not isomorphic to the group of symmetries of a square.

\hfill \mbox{\textit{OCR MEI FP3  Q4}}