Transpositions of [n]

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $n\in\mathbb{N}$, with $[n]$ as in Intervals of Natural Numbers §segment and $S_{n}$ as in Permutations of a Set and the Set of Permutations of [n] §permutations.

¶ For $p,q\in[n]$ with $p\neq q$, the transposition of $p$ and $q$ is the map $\theta_{pq}\in S_{n}$ of Permutations of [n]: Identity, Composition, Inverses, Translations and Transpositions §transposition; thus $\theta_{pq}(p)=q$, $\theta_{pq}(q)=p$, and $\theta_{pq}(k)=k$ for every $k\in[n]$ with $k\neq p$ and $k\neq q$.