Rothaus–Thompson conjugacy-class tiling problem

Rothaus and Thompson posed the relevant tiling question in 1966, asking whether the symmetric group can be tiled by the identity together with all transpositions. The broader conjugacy-class problem remains focused on the exceptional transposition case.

Known results

• Rothaus–Thompson, 1966: nonexistence when $1+n(n-1)/2$ has a prime factor $p\geq\sqrt{n}+2$.
• Nomura, 1985: related necessary conditions for the transposition tiling problem.
• Somlai, Xia, and Zhou, 2025: strengthened the obstruction to $p\geq\sqrt{n}+1$ and derived partition-transitivity conditions on a complementary set.
• The cases involving $T_n=\{\mathrm{id}\}\cup\{\text{all transpositions}\}$ and the transposition class $T_n^*$ remain conjectural for general $n\geq4$.

October 2026 preprint

Somlai, Xia, and Zhou’s reported preprint claims to eliminate every symmetric-group conjugacy class except transpositions and to settle the corresponding alternating-group statement in a stronger normal-subset form. The symmetric-group transposition case remains open; the claimed advance has not received independent mathematical assessment in the retrieved sources.

Current status (as of October 2026): The general symmetric-group problem is reduced, according to an unverified preprint, to the transposition case, while the claimed alternating-group analogue is reported as settled; the transposition case itself remains open.