Abstract: Classical theorems of Engel and Zorn describe the classes of finite dimensional nilpotent Lie algebras and finite groups in terms of two-variable identities. Recently similar characterizations have been obtained for the classes of finite dimensional solvable Lie algebras and finite solvable groups (the proof of the latter one required a good bunch of arithmetic geometry and computer algebra).
More generally, a theorem of Baer describes the nilpotent radical of a finite group in terms of Engel elements. Our goal is to obtain similar characterizations for the solvable radical of a finite dimensional Lie algebra and of a finite group.
This talk is based on several (finished as well as ongoing) projects joint with T.Bandman, M.Borovoi, N.Gordeev, G.-M.Greuel, F.Grunewald, D.Nikolova, G.Pfister, E.Plotkin, and A.Shalev.