4 The twelve distinct elements of an abelian multiplicative group \(G\) are $$e , a , a ^ { 2 } , a ^ { 3 } , a ^ { 4 } , a ^ { 5 } , b , a b , a ^ { 2 } b , a ^ { 3 } b , a ^ { 4 } b , a ^ { 5 } b$$ where \(e\) is the identity element, \(a ^ { 6 } = e\) and \(b ^ { 2 } = e\).

1. Show that the element \(a ^ { 2 } b\) has order 6 .
2. Show that \(\left\{ e , a ^ { 3 } , b , a ^ { 3 } b \right\}\) is a subgroup of \(G\).
3. List all the cyclic subgroups of \(G\). You are given that the set $$H = \{ 1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59,61,67,71,73,77,79,83,89 \}$$ with binary operation multiplication modulo 90 is a group.
4. Determine the order of each of the elements 11, 17 and 19 .
5. Give a cyclic subgroup of \(H\) with order 4.
6. By identifying possible values for the elements \(a\) and \(b\) above, or otherwise, give one example of each of the following:
   (A) a non-cyclic subgroup of \(H\) with order 12,
   (B) a non-cyclic subgroup of \(H\) with order 4.