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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?

Jamāl
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Peter Liu
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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.

Robert Lewis
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    Thanks, but what does "|A| letters" mean? – Peter Liu Mar 09 '20 at 10:14
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    @Peter: The cardinality of $A$; when $A$ is a finite set, it is the number of element in $A$. – Arturo Magidin Mar 09 '20 at 18:54
  • @Peter: to refine Arturo Magidin's comment one should properly say that the phrase :$\vert A \vert$ letters" refers to the the $\vert A \vert$ elements of a set on which $S_A$ acts; they are often represented by $\vert A \vert$ symbols $s_1, s_2, \ldots, s_{\vert A \vert}$ which are called the "letters", since the form a sort of "alphabet", though perhaps for no language other than a mathematical one!. – Robert Lewis Mar 09 '20 at 20:01
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    Thanks, your explanation is very clear! – Peter Liu Mar 10 '20 at 00:02
  • @Peter: my pleasure, sir, and thanks for the "acceptance"! Cheers! – Robert Lewis Mar 10 '20 at 00:05