8 A set of matrices \(M\) is defined by
$$A = \left( \begin{array} { l l }
1 & 0
0 & 1
\end{array} \right) , \quad B = \left( \begin{array} { c c }
\omega & 0
0 & \omega ^ { 2 }
\end{array} \right) , \quad C = \left( \begin{array} { c c }
\omega ^ { 2 } & 0
0 & \omega
\end{array} \right) , \quad D = \left( \begin{array} { c c }
0 & 1
1 & 0
\end{array} \right) , \quad E = \left( \begin{array} { c c }
0 & \omega ^ { 2 }
\omega & 0
\end{array} \right) , \quad F = \left( \begin{array} { c c }
0 & \omega
\omega ^ { 2 } & 0
\end{array} \right) ,$$
where \(\omega\) and \(\omega ^ { 2 }\) are the complex cube roots of 1 . It is given that \(M\) is a group under matrix multiplication.
- Write down the elements of a subgroup of order 2.
- Explain why there is no element \(X\) of the group, other than \(A\), which satisfies the equation \(X ^ { 5 } = A\).
- By finding \(B E\) and \(E B\), verify the closure property for the pair of elements \(B\) and \(E\).
- Find the inverses of \(B\) and \(E\).
- Determine whether the group \(M\) is isomorphic to the group \(N\) which is defined as the set of numbers \(\{ 1,2,4,8,7,5 \}\) under multiplication modulo 9 . Justify your answer clearly.