Wilson conjecture for omega-categorical Lie algebras, the case 3-Engel characteristic 5

Abstract

We prove a version of the Wilson conjecture for $\omega$-categorical $3$-Engel Lie algebras over a field of characteristic $5$: every $\omega$-categorical Lie algebra over $\mathbb{F}_5$ which satisfies the identity $[x,y^3] = 0$ is nilpotent. We also include an extended introduction to Wilson's conjecture: \textit{every $\omega$-categorical locally nilpotent $p$-group is nilpotent}, and present variants of this conjecture and connections to local/global nilpotency problems (Burnside, Kurosh-Levitzki, Engel groups). No particular knowledge of model theory is assumed except basic notions of formulas and definable sets.