Wilson’s conjecture on locally nilpotent countably categorical groups

Wilson’s 1981 conjecture asserts that every locally nilpotent countably categorical group is nilpotent.

Known results

• Countably categorical $3$-Engel groups are nilpotent.
• Countably categorical $4$-Engel groups of exponent $5$ are nilpotent.
• Countably categorical $4$-Engel groups of odd exponent are nilpotent.
• Related Lie-algebra cases were established in characteristics $3$ and $5$, with corresponding group-theoretic consequences via the Lazard correspondence.

August 2026 Engel-group preprint

Christian d’Elbée’s preprint uses Lie methods and a computer-algebra-checked exceptional Lazard-correspondence case to prove the listed Engel-group results. These are the first nontrivial Engel-group cases reported for Wilson’s conjecture, not a proof of the general statement.

Current status (as of August 2026): the $3$-Engel case and several $4$-Engel exponent cases are proved in a preprint, while the general conjecture remains open.