Transpositions of adjacent elements generate the symmetric group on a finite set

Statement

Let $S$ be a totally ordered finite set. Then, the set of Transposition (?)s between adjacent elements in the total ordering of $S$ generates the Symmetric group (?) on $S$.

Note that any totally ordered set can be identified with the set $\{1,2,3,\dots ,n\}$ with the usual integer ordering, so the above can be rephrased as: the set of transpositions of the form $(i,i+1)$ generate the symmetric group on $\{1,2,3,\dots ,n\}$.

It turns out that this generating set is both Sims-reduced and Jerrum-reduced.

Related facts

• All transpositions involving one element generate the finitary symmetric group

Proof

The bubble sort proof

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