A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y ∈ G such that for any x ∈ G the nth commutator [x, y, . . . , y] equals 1 for n big enough. We obtain a characterization of the solvable radical of a finite dimensional Lie algebra defined over a field of characteristic zero in similar terms. We suggest a conjectu...

Engel groups and Engel elements became popular in 50s. We consider in the paper the more general nil-groups and nil-elements in groups. All these notions are related to nilpotent groups and nilpo-tent radicals in groups. These notions generate problems which are parallel to Burnside problems for periodic groups. The first three theorems of the paper are devoted to nil-groups and Engel groups, w...

We give an infinite family of examples that generalize the construction given by Noce, Tracey and Traustason a locally finite 2-group G containing left 3-Engel element x where 〈x〉G, normal closure in G, is not nilpotent. The based on Lie algebras are interest their own right make use classical theorem Lucas, regarding when (mn) even.