Transpositions generate the finitary symmetric group
Statement
For a finite set
The symmetric group on a finite set is generated by the transpositions in it. A transposition is a permutation that interchanges two elements and leaves all the others fixed; in other words, it is a cycle of length two.
For an infinite set
The finitary symmetric group on any set is generated by the transpositions in it. A transposition is a permutation that interchanges two elements and leaves all the others fixed; in other words, it is a cycle of length two.
Related facts
In terms of IAPS theory
In terms of IAPS theory, this translates to saying that the transposition on a two-element set forms a one-element permutatively generating set for the permutation IAPS.