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

TL;DR

This paper proves a version of Wilson's conjecture for omega-categorical 3-Engel Lie algebras over a field of characteristic 5, showing such algebras satisfying a specific identity are nilpotent.

Contribution

It establishes that omega-categorical 3-Engel Lie algebras over characteristic 5 fields satisfying a certain identity are necessarily nilpotent, advancing understanding of the structure of these algebras.

Findings

Omega-categorical 3-Engel Lie algebras over characteristic 5 fields are nilpotent if they satisfy [x,y^3]=0.

The paper extends Wilson's conjecture to a specific algebraic context.

Connections between local nilpotency and global nilpotency are discussed.

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.