When is $F(A)$ an abelian group?

Question

Let $A$ be a nonempty finite set. Denote by $F(A)$ the set of all bijections $f : A \to A$. When is $F(A)$ an abelian group?

Answer 1

A bijection

$f: A \longleftrightarrow A \tag 1$

is essentially a permutation of the elements of $A$; thus it may be construed as a member of $S_{\vert A \vert }$, the symmetric group acting on $\vert A \vert$ letters. This group is not abelian for $\vert A \vert \ge 3$, so the bijections don't form an abelian group unless $\vert A \vert \le 2$.

For $\vert A \vert = 1$, $F(A)$ only contains the identity mapping; when $\vert A \vert = 2$, we have the identity and the map which swaps the two elements of $A$; indeed, when $\vert A \vert = 2$, $F(A)$ is $\Bbb Z_2$, the unique two-element group; it is easy to see these commute. So for $\vert A \vert \le 2$, $F(A)$ is abelian.