\(\mathbf { 4 }\) 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 $$\mathbf { I } = \left( \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right) , \quad \mathbf { J } = \left( \begin{array} { r r } \mathrm { j } & 0
0 & - \mathrm { j } \end{array} \right) , \quad \mathbf { K } = \left( \begin{array} { r r } 0 & 1
- 1 & 0 \end{array} \right) , \quad \mathbf { L } = \left( \begin{array} { c c } 0 & \mathrm { j }
\mathrm { j } & 0 \end{array} \right)$$

1. Copy and complete the following composition table for \(G\).

   	I	J	K	L	-I	-J	-K	\(- \mathbf { L }\)
   I	I	J	K	L	-I	-J	-K	-L
   J	J	-I	L	-K	-J	I	-L	K
   K	K	-L	-I
   L	L	K
   -I	-I	-J
   -J	-J	I
   -K	-K	L
   -L	-L	-K

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