Tiling the symmetric group by transpositions

Abstract For nonempty subsets and of a group , we say that is a tiling of if every element of can be uniquely expressed as for some and . In 1966, Rothaus and Thompson studied whether the symmetric group with admits a tiling , where consists of the identity and all the transpositions in . They showed that no such tiling exists if is divisible by a prime number at least . In this paper, we establish a new necessary condition for the existence of such a tiling: the subset must be partition‐transitive with respect to certain partitions of . This generalizes the result of Rothaus and Thompson, as well as a result of Nomura in 1985. We also study whether can be tiled by the set of all the transpositions, which finally leads us to conjecture that neither nor tiles for any .