The Sylow–Burnside process is a Markov chain on \(S_n\) which can be used to uniformly sample Sylow \(p\) -double cosets. This article implements and analyzes the Sylow–Burnside process in the case when \(n = pk\) with \(p\) prime and \(k < p\) . The main result is the limiting profile of the distance to stationarity for the Sylow–Burnside process as \(p\) goes to infinity. A corollary of the limiting profile is that the mixing time of the Sylow–Burnside process is on the order of \(p\log k\) and cutoff occurs if and only if \(k \rightarrow \infty \) as \(p \rightarrow \infty \) . The limit profile is derived by first showing that the Sylow–Burnside process lumps to the size of the double coset and then approximating the lumped process. The distance to stationarity for the approximating process can be computed exactly. This computation gives explicit upper and lower bounds on the distance to stationarity for the Sylow–Burnside process. These non-asymptotic bounds give very accurate approximations even for \(p\) as small as 11.
