Chapter 3.1, Problem 46E

In Example $3$, the group $S(A)$ is nonabelian where $A=\left\{1,2,3\right\}$. Exhibit a set $A$ such that $S(A)$ is abelian.

Example 3.

We shall take $A=\left\{1,2,3\right\}$ 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 once choice for $f(3)$. Hence there are $3!=3\cdot 2\cdot 1$ different mappings $f$ in $S(A)$.