Rothaus–Thompson transposition tiling problem

Determine whether the conjugacy class of transpositions tiles the symmetric group S_n; equivalently, establish whether the conjugacy class of transpositions fails to tile S_n for every n≥4, as conjectured by Fang and Xia.

Background

The paper studies when a conjugacy class of a finite symmetric or alternating group can tile the group. For symmetric groups, the authors prove that every nonidentity conjugacy class other than the class of transpositions does not tile S_n. Consequently, the transposition class is the sole unresolved case of the generalized Rothaus–Thompson problem.

The transposition case was first investigated by Rothaus and Thompson in 1966. Subsequent work established necessary conditions for a tiling complement, but did not resolve existence. Fang and Xia conjectured that, for every n≥4, the transposition class does not tile S_n.