7 You are given the set \(S = \{ 1,5,7,11,13,17 \}\) together with \(\times _ { 18 }\), the operation of multiplication modulo 18.
- Complete the Cayley table for \(\left( S , \times _ { 18 } \right)\) given in the Printed Answer Booklet.
- Prove that ( \(S , \times _ { 18 }\) ) is a group. (You may assume that \(\times _ { 18 }\) is associative.)
- Write down the order of each element of the group.
- Show that \(\left( S , \times _ { 18 } \right)\) is a cyclic group.
- Give an example of a non-cyclic group of order 6 .
- Give one reason why your example is structurally different to \(\left( S , { } _ { 18 } \right)\).