Matrix groups

7 The set \(M\) consists of the six matrices \(\left( \begin{array} { l l } 1 & 0 \\ n & 1 \end{array} \right)\), where \(n \in \{ 0,1,2,3,4,5 \}\). It is given that \(M\) forms a group ( \(M , \times\) ) under matrix multiplication, with numerical addition and multiplication both being carried out modulo 6 .

1. Determine whether ( \(M , \times\) ) is a commutative group, justifying your answer.
2. Write down the identity element of the group and find the inverse of \(\left( \begin{array} { l l } 1 & 0 \\ 2 & 1 \end{array} \right)\).
3. State the order of \(\left( \begin{array} { l l } 1 & 0 \\ 3 & 1 \end{array} \right)\) and give a reason why \(( M , \times )\) has no subgroup of order 4.
4. The multiplicative group \(G\) has order 6. All the elements of \(G\), apart from the identity, have order 2 or 3 . Determine whether \(G\) is isomorphic to ( \(M , \times\) ), justifying your answer.

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.

1. Write down the elements of a subgroup of order 2.
2. Explain why there is no element \(X\) of the group, other than \(A\), which satisfies the equation \(X ^ { 5 } = A\).
3. By finding \(B E\) and \(E B\), verify the closure property for the pair of elements \(B\) and \(E\).
4. Find the inverses of \(B\) and \(E\).
5. 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.

4 The group \(G\) consists of a set of six matrices under matrix multiplication. Two of the elements of \(G\) are \(\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 1 & - 1 \\ 0 & - 1 \end{array} \right)\).

1. Determine each of the following:
   • \(\mathbf { A } ^ { 2 }\)
   • \(\mathbf { B } ^ { 2 }\)
   • Determine all the elements of \(G\).
   • State the order of each non-identity element of \(G\).
   • State, with justification, whether \(G\) is
   • abelian
   • cyclic.