5 A multiplicative group \(G\) of order 9 has distinct elements \(p\) and \(q\), both of which have order 3 . The group is commutative, the identity element is \(e\), and it is given that \(q \neq p ^ { 2 }\).
- Write down the elements of a proper subgroup of \(G\)
(a) which does not contain \(q\),
(b) which does not contain \(p\). - Find the order of each of the elements \(p q\) and \(p q ^ { 2 }\), justifying your answers.
- State the possible order(s) of proper subgroups of \(G\).
- Find two proper subgroups of \(G\) which are distinct from those in part (i), simplifying the elements.