9 The set \(S\) consists of the numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\). ( \(\mathbb { Z }\) denotes the set of integers \(\{ 0 , \pm 1 , \pm 2 , \ldots \}\).)

1. Prove that the elements of \(S\), under multiplication, form a commutative group \(G\). (You may assume that addition of integers is associative and commutative.)
2. Determine whether or not each of the following subsets of \(S\), under multiplication, forms a subgroup of \(G\), justifying your answers.
   (a) The numbers \(3 ^ { 2 n }\), where \(n \in \mathbb { Z }\).
   (b) The numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\) and \(n \geqslant 0\).
   (c) The numbers \(3 ^ { \left( \pm n ^ { 2 } \right) }\), where \(n \in \mathbb { Z }\). 1 (a) A group \(G\) of order 6 has the combination table shown below. \(G\) and \(H\) are the non-cyclic groups of order 4 and 6 respectively.
3. Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups \(G\) and \(H\). Hence explain why there are no non-cyclic proper subgroups of \(Q\). 7 Three planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) have equations $$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 5 , \quad \mathbf { r } . ( \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 6 , \quad \mathbf { r } . ( \mathbf { i } + 5 \mathbf { j } - 12 \mathbf { k } ) = 12 ,$$ respectively. Planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in a line \(l\); planes \(\Pi _ { 2 }\) and \(\Pi _ { 3 }\) intersect in a line \(m\).
4. Show that \(l\) and \(m\) are in the same direction.
5. Write down what you can deduce about the line of intersection of planes \(\Pi _ { 1 }\) and \(\Pi _ { 3 }\).
6. By considering the cartesian equations of \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\), or otherwise, determine whether or not the three planes have a common line of intersection. 8 The operation \(*\) is defined on the elements \(( x , y )\), where \(x , y \in \mathbb { R }\), by $$( a , b ) * ( c , d ) = ( a c , a d + b ) .$$ It is given that the identity element is \(( 1,0 )\).
7. Prove that \(*\) is associative.
8. Find all the elements which commute with \(( 1,1 )\).
9. It is given that the particular element \(( m , n )\) has an inverse denoted by \(( p , q )\), where $$( m , n ) * ( p , q ) = ( p , q ) * ( m , n ) = ( 1,0 ) .$$ Find \(( p , q )\) in terms of \(m\) and \(n\).
10. Find all self-inverse elements.
11. Give a reason why the elements \(( x , y )\), under the operation \(*\), do not form a group.