1. The set of matrices \(G = \{ \mathbf { I } , \mathbf { A } , \mathbf { B } , \mathbf { C } , \mathbf { D } , \mathbf { E } \}\) where

$$\mathbf { I } = \left( \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right) \quad \mathbf { A } = \left( \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { l l } 1 & 1
1 & 0 \end{array} \right) \quad \mathbf { C } = \left( \begin{array} { l l } 1 & 1
0 & 1 \end{array} \right) \quad \mathbf { D } = \left( \begin{array} { l l } 1 & 0
1 & 1 \end{array} \right) \quad \mathbf { E } = \left( \begin{array} { l l } 0 & 1
1 & 1 \end{array} \right)$$ with the operation \(\otimes _ { 2 }\) of matrix multiplication with entries evaluated modulo 2 , forms a group.

1. Show that \(\mathbf { B }\) is an element of order 3 in \(G\).
2. Determine the orders of the other elements of \(G\).
3. Give a reason why \(G\) is not isomorphic to
   1. a cyclic group of order 6
   2. the group of symmetries of a regular hexagon. The group \(H\) of permutations of the numbers 1, 2 and 3 contains the following elements, denoted in two-line notation, $$\begin{array} { l l l } e = \left( \begin{array} { l l l } 1 & 2 & 3
      1 & 2 & 3 \end{array} \right) & a = \left( \begin{array} { l l l } 1 & 2 & 3
      2 & 3 & 1 \end{array} \right) & b = \left( \begin{array} { l l l } 1 & 2 & 3
      3 & 1 & 2 \end{array} \right)
      c = \left( \begin{array} { l l } 1 & 2
      1 & 3
      2 \end{array} \right) & d = \left( \begin{array} { l l l } 1 & 2 & 3
      2 & 1 & 3 \end{array} \right) & f = \left( \begin{array} { l l } 1 & 2
      3 & 2 \end{array} \right) \end{array}$$
4. Determine an isomorphism between the groups \(G\) and \(H\).