Kourovka Notebook Question 18.48

Question 18.48 asks whether only finitely many orders occur among products of two class transpositions. Earlier work settled important subclasses, while a new preprint claims the full answer is affirmative.

Known results

• Bardakov and Iskra (2024): for horizontal class transpositions, the possible orders are exactly $\{1,2,3,4,6,12\}$, with examples for every value.
• Bardakov and Iskra: computationally observed general orders all divide $840$, but this was not proved for arbitrary class transpositions.
• A subsequent paper treats slanted class transpositions only under specified conditions and does not establish the general result.

October 2026 claimed resolution

Alex Iskra’s preprint Finite Orders of Products of Two Class Transpositions Divide 840 claims both the divisibility bound by $840$ and a sharpness construction, hence an affirmative answer. The claim is unrefereed and has not received independent mathematical assessment in the retrieved sources.

Current status (as of October 2026): A preprint claims the general question is solved, but the claim remains unverified; the earlier subclass results are established.