Transposition (as an element of a symmetric group)

by Patrick Stevens Jun 15 2016

A transposition is the simplest kind of permutation: it swaps two elements.

In a Symmetric group, a transposition is a permutation which has the effect of swapping two elements while leaving everything else unchanged. More formally, it is a permutation of order $~$2$~$ which fixes all but two elements.

%%%knows-requisite(Cycle type of a permutation): A transposition is precisely an element with cycle type $~$2$~$. %%%


In $~$S_5$~$, the permutation $~$(12)$~$ is a transposition: it swaps $~$1$~$ and $~$2$~$ while leaving all three of the elements $~$3,4,5$~$ unchanged. However, the permutation $~$(124)$~$ is not a transposition, because it has order $~$3$~$, not order $~$2$~$.