4 The group \(F = \{ \mathrm { p } , \mathrm { q } , \mathrm { r } , \mathrm { s } , \mathrm { t } , \mathrm { u } \}\) consists of the six functions defined by
$$\mathrm { p } ( x ) = x \quad \mathrm { q } ( x ) = 1 - x \quad \mathrm { r } ( x ) = \frac { 1 } { x } \quad \mathrm {~s} ( x ) = \frac { x - 1 } { x } \quad \mathrm { t } ( x ) = \frac { x } { x - 1 } \quad \mathrm { u } ( x ) = \frac { 1 } { 1 - x } ,$$
the binary operation being composition of functions.
- Show that st \(= \mathrm { r }\) and find ts.
- Copy and complete the following composition table for \(F\).
| p | q | r | s | t | u |
| p | p | q | r | s | t | u |
| q | q | p | s | r | u | t |
| r | r | u | p | t | s | q |
| s | s | t | q | u | r | p |
| t | t | s | u | | | |
| u | u | r | t | | | |
- Give the inverse of each element of \(F\).
- List all the subgroups of \(F\).
The group \(M\) consists of \(\left\{ 1 , - 1 , e ^ { \frac { \pi } { 3 } \mathrm { j } } , e ^ { - \frac { \pi } { 3 } \mathrm { j } } , e ^ { \frac { 2 \pi } { 3 } \mathrm { j } } , e ^ { - \frac { 2 \pi } { 3 } \mathrm { j } } \right\}\) with multiplication of complex numbers as its binary operation.
- Find the order of each element of \(M\).
The group \(G\) consists of the positive integers between 1 and 18 inclusive, under multiplication modulo 19.
- Show that \(G\) is a cyclic group which can be generated by the element 2 .
- Explain why \(G\) has no subgroup which is isomorphic to \(F\).
- Find a subgroup of \(G\) which is isomorphic to \(M\).