Tiling symmetric groups by transpositions

For two nonempty subsets X and Y of a group G, we say that $(X,Y)$ is a tiling of G if every element of G can be uniquely expressed as $xy$ for some x∈ X and y∈ Y. In 1966, Rothaus and Thompson studied whether the symmetric group Sₙ with n≥3 admits a tiling (Tₙ,Y), where Tₙ consists of the identity and all the transpositions in Sₙ. They showed that no such tiling exists if $1+n(n-1)/2$ is divisible by a prime number at least √n+2. In this paper, we establish a new necessary condition for the existence of such a tiling: the subset Y must be partition-transitive with respect to certain partitions of n. This generalizes the result of Rothaus and Thompson, as well as a result of Nomura in 1985. We also study whether Sₙ can be tiled by the set Tₙ^* of all transpositions, which finally leads us to conjecture that neither Tₙ nor Tₙ^* tiles Sₙ for any n≥3.