  This paper explores the mixing time of the random transposition walk on the
symmetric group. While it has long been known that this walk mixes in order
n*log(n) time, this result has not previously been attained using coupling. A
coupling argument showing the correct order mixing time is presented. This is
accomplished by first projecting to conjugacy classes, and then using the
Bubley-Dyer path coupling construction. In order to obtain appropriate bounds
on the time it takes the path coupling to meet, ideas from Schramm's paper
"Compositions of Random Transpositions" are used.
