We consider decidability problems associated with Engel’s identity ([· · · [[x, y], y], . . . , y] = 1 for a long enough commutator sequence) in groups generated by an automaton. We give a partial algorithm that decides, given x, y, whether an Engel identity is satisfied. It succeeds, importantly, in proving that Grigorchuk’s 2-group is not Engel. We consider next the problem of recognizing Eng...

We consider decidability problems in self-similar semigroups, and in particular in semigroups of automatic transformations of X. We describe algorithms answering the word problem, and bound its complexity under some additional assumptions. We give a partial algorithm that decides in a group generated by an automaton, given x, y, whether an Engel identity (r ̈ ̈ ̈ rrx, ys, ys, . . . , ys “ 1 for ...

Let G be a non-Engel group and let L(G) be the set of all left Engel elements of G. Associate with G a graph EG as follows: Take G\L(G) as vertices of EG and join two distinct vertices x and y whenever [x,k y] 6= 1 and [y,k x] 6= 1 for all positive integers k. We call EG, the Engel graph of G. In this paper we study the graph theoretical properties of EG.

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...