- A group \(G\) contains distinct elements \(a , b\) and \(e\) where \(e\) is the identity element and the group operation is multiplication.
Given \(a ^ { 2 } b = b a\), prove \(a b \neq b a\)
(ii) The set \(H = \{ 1,2,4,7,8,11,13,14 \}\) forms a group under the operation of multiplication modulo 15
- Find the order of each element of \(H\).
- Find three subgroups of \(H\) each of order 4, and describe each of these subgroups.
The elements of another group \(J\) are the matrices \(\left( \begin{array} { c c } \cos \left( \frac { k \pi } { 4 } \right) & \sin \left( \frac { k \pi } { 4 } \right)
- \sin \left( \frac { k \pi } { 4 } \right) & \cos \left( \frac { k \pi } { 4 } \right) \end{array} \right)\)
where \(k = 1,2,3,4,5,6,7,8\) and the group operation is matrix multiplication. - Determine whether \(H\) and \(J\) are isomorphic, giving a reason for your answer.