5 The group \(G\) consists of a set \(S\) together with \(\times _ { 80 }\), the operation of multiplication modulo 80. It is given that \(S\) is the smallest set which contains the element 11 .
- By constructing the Cayley table for \(G\), determine all the elements of \(S\).
The Cayley table for a second group, \(H\), also with the operation \(\times _ { 80 }\), is shown below.
| \cline { 2 - 5 }
\multicolumn{1}{c|}{\(\times _ { 80 }\)} | 1 | 9 | 31 | 39 |
| 1 | 1 | 9 | 31 | 39 |
| 9 | 9 | 1 | 39 | 31 |
| 31 | 31 | 39 | 1 | 9 |
| 39 | 39 | 31 | 9 | 1 |
- Use the two Cayley tables to explain why \(G\) and \(H\) are not isomorphic.
- List
- all the proper subgroups of \(G\),
- all the proper subgroups of \(H\).
(ii) Use your answers to (c) (i) to give another reason why \(G\) and \(H\) are not isomorphic.