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.

1. Show that st \(= \mathrm { r }\) and find ts.
2. 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

3. Give the inverse of each element of \(F\).
4. 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.
5. 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.
6. Show that \(G\) is a cyclic group which can be generated by the element 2 .
7. Explain why \(G\) has no subgroup which is isomorphic to \(F\).
8. Find a subgroup of \(G\) which is isomorphic to \(M\).