Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 5 / a / ii / Solution

For a singleton $X$, the symmetric group contains only the identity function. For a two-element set, it contains the identity function and the transposition exchanging the two elements; the latter squares to the identity function. In both cases all elements commute, so the group is Abelian.

If $X$ contains three distinct elements $a,b,c$, define the transpositions $\sigma =(ab)$ and $\tau =(bc)$, fixing every other element of $X$. They are bijections, even when $X$ is infinite. With function composition read from right to left,Thus they do not commute, so the symmetric group is not Abelian. Since $X$ is nonempty, this proves