5 The set \(S\) is defined as $$S = \{ A , B , C , D \}$$ where
\(A = \left[ \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right] \quad B = \left[ \begin{array} { c c } 0 & - 1
1 & 0 \end{array} \right] \quad C = \left[ \begin{array} { c c } - 1 & 0
0 & - 1 \end{array} \right] \quad D = \left[ \begin{array} { c c } 0 & 1
- 1 & 0 \end{array} \right]\) The group \(G\) is formed by \(S\) under matrix multiplication.
The group \(H\) is defined as \(H = ( \langle \mathrm { i } \rangle , \times )\), where \(\mathrm { i } ^ { 2 } = - 1\)
5

1. 1. Prove that \(B\) is a generator of \(G\).
      Fully justify your answer.
      5
2. (ii) Show that \(G \cong H\).
   Fully justify your answer.
   5
3. 1. Explain why \(H\) has no subgroups of order 3
      Fully justify your answer.
      5
4. (ii) Find all of the subgroups of \(H\).