4 The group \(G = \{ 1,2,3,4,5,6 \}\) has multiplication modulo 7 as its operation. The group \(H = \{ 1,5,7,11,13,17 \}\) has multiplication modulo 18 as its operation.

1. Show that the groups \(G\) and \(H\) are both cyclic.
2. List all the proper subgroups of \(G\).
3. Specify an isomorphism between \(G\) and \(H\). The group \(S = \{ \mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d } , \mathrm { e } , \mathrm { f } \}\) consists of functions with domain \(\{ 1,2,3 \}\) given by $$\begin{array} { l l l } \mathrm { a } ( 1 ) = 2 & \mathrm { a } ( 2 ) = 3 & \mathrm { a } ( 3 ) = 1
   \mathrm {~b} ( 1 ) = 3 & \mathrm {~b} ( 2 ) = 1 & \mathrm {~b} ( 3 ) = 2
   \mathrm { c } ( 1 ) = 1 & \mathrm { c } ( 2 ) = 3 & \mathrm { c } ( 3 ) = 2
   \mathrm {~d} ( 1 ) = 3 & \mathrm {~d} ( 2 ) = 2 & \mathrm {~d} ( 3 ) = 1
   \mathrm { e } ( 1 ) = 1 & \mathrm { e } ( 2 ) = 2 & \mathrm { e } ( 3 ) = 3
   \mathrm { f } ( 1 ) = 2 & \mathrm { f } ( 2 ) = 1 & \mathrm { f } ( 3 ) = 3 \end{array}$$ and the group operation is composition of functions.
4. Show that ad \(= \mathrm { c }\) and find da.
5. Give a reason why \(S\) is not isomorphic to \(G\).
6. Find the order of each element of \(S\).
7. List all the proper subgroups of \(S\).