4 A binary operation * is defined on real numbers \(x\) and \(y\) by $$x * y = 2 x y + x + y$$ You may assume that the operation \(*\) is commutative and associative.

1. Explain briefly the meanings of the terms 'commutative' and 'associative'.
2. Show that \(x * y = 2 \left( x + \frac { 1 } { 2 } \right) \left( y + \frac { 1 } { 2 } \right) - \frac { 1 } { 2 }\). The set \(S\) consists of all real numbers greater than \(- \frac { 1 } { 2 }\).
3. (A) Use the result in part (ii) to show that \(S\) is closed under the operation *.
   (B) Show that \(S\), with the operation \(*\), is a group.
4. Show that \(S\) contains no element of order 2 . The group \(G = \{ 0,1,2,4,5,6 \}\) has binary operation ∘ defined by $$x \circ y \text { is the remainder when } x * y \text { is divided by } 7 \text {. }$$
5. Show that \(4 \circ 6 = 2\). The composition table for \(G\) is as follows.

   \(\circ\)	0	1	2	4	5	6
   0	0	1	2	4	5	6
   1	1	4	0	6	2	5
   2	2	0	5	1	6	4
   4	4	6	1	5	0	2
   5	5	2	6	0	4	1
   6	6	5	4	2	1	0

6. Find the order of each element of \(G\).
7. List all the subgroups of \(G\).