Chapter 3.4, Problem 1E

List all cyclic subgroups of the group $S(A)$ in Example $3$ of section $3.1$.

Example 3.

We shall take $A=\{1,2,3\}$ and obtain an explicit example of $S(A)$. In order to define an element $f$ of $S(A)$, we need to specify $f(1)$, $f(2)$, and $f(3)$. There are three possible choices for $f(1)$. Since $f$ is to be bijective, there are two choices for $f(2)$ after $f(1)$ has been designated, and then only one choice for $f(3)$. Hence there are $3!=3\cdot 2\cdot 1$ different mappings $f$ in $S(A)$.