Non-tiling conjecture for transposition classes

Prove that, for every n≥4, the conjugacy class of transpositions does not tile the symmetric group S_n.

Background

The unresolved transposition case is the only case not settled by the paper’s classification of conjugacy classes of S_n. A tiling by the transposition class would require a subset of S_n serving as a unique factorization complement.

Fang and Xia formulated the specific conjecture that neither the transposition class nor the closely related set obtained by adjoining the identity tiles S_n for n≥4. The present paper leaves the transposition-class case unresolved and proves non-tiling for all other nonidentity conjugacy classes.

References

Despite subsequent work (see), this Rothaus--Thompson problem remains open. In fact, Fang and Xia conjectured that, for $n\geq4$, the conjugacy class of transpositions does not tile $S_n$.

— Tilings of symmetric and alternating groups by conjugacy classes  (2610.01267 - Somlai et al., 1 Oct 2026) in Section 1, Introduction