8 A multiplicative group \(H\) has the elements \(\left\{ e , a , a ^ { 2 } , a ^ { 3 } , w , a w , a ^ { 2 } w , a ^ { 3 } w \right\}\) where \(e\) is the identity, elements \(a\) and \(w\) have orders 4 and 2 respectively and \(w a = a ^ { 3 } w\).
- Show that \(w a ^ { 2 } = a ^ { 2 } w\) and also that \(w a ^ { 3 } = a w\).
- Hence show that each of \(a w , a ^ { 2 } w\) and \(a ^ { 3 } w\) has order 2 .
- Find two non-cyclic subgroups of \(H\) of order 4, and show that they are not cyclic.