Draw the Cayley table for $\left\{ {G,\left.  *  \right\}} \right.$ .

(i)     Determine the order of each element of $\left\{ {G,\left.  *  \right\}} \right.$ .

(ii)     Find all the proper subgroups of $\left\{ {G,\left.  *  \right\}} \right.$ .

Solve the equation $x * 6 * x = 3$ where $x \in G$ .

     A3

Note: Award A2 for 1 error, A1 for 2 errors, A0 for 3 or more errors.

[3 marks]

(i)     We first identify $1$ as the identity     (A1)

Order of $1 = 1$

Order of $2 = 3$

Order of $3 = 6$

Order of $4 = 3$

Order of $5 = 6$

Order of $6 = 2$     A3

Note: Award A2 for 1 error, A1 for 2 errors, A0 for more than 2 errors.

(ii)     $\left\{ {1,\left. 6 \right\}} \right.$ ; $\left\{ {1,\left. {2,4} \right\}} \right.$     A1A1

[6 marks]

The equation is equivalent to

$6 * x * x = 3$     M1

$x * x = 4$

$x = 2$ or $5$     A1A1

[3 marks]