Engel lattice-ordered groups

An Engel l-group generating a proper normal-valued l-variety is shown to be o-approximable. It is also established that for every proper normal-valued l-varietyF, the class (F) of Engel l-groups from F is a torsion class.Translated fromAlgebra i Logika, Vol. 34, No. 4, pp. 398–404, July-August, 1995.     

Engel groups I

In this series of papers we study Engel groups through their presentation, using Small Cancellation Theory. In the present paper we describe the consequences of a single Engel relator.     

Engel groups III

Let F denote a free group. An n-Engel word of F is an iterated commutator of the form [A, nB], where (A,B) is a pair of elements of F, which we here assume to generate a non-cyclic subgroup of F. This is the second of a series of three papers that examine some of the properties of a group defined by a suitable set of relators of the form [A _i , nB _i ], i = 1, 2, ...,m (where n is fixed). In the present paper, the techniques of [1] are generalised in a way that permits the application of the conclusions of [1] concerning the nature of the consequences of a single Engel relator to the case of a finite number of n-Engel relators, subject only to some mild hypotheses on the lengths of the relators and the value of n.     

Bounded Engel elements in groups satisfying an identity

We prove that a residually finite group G satisfying an identity \(w\equiv 1\) and generated by a commutator closed set X of bounded left Engel elements is locally nilpotent. We also extend such a result to locally graded groups, provided that X is a normal set. As an immediate consequence, we obtain that a locally graded group satisfying an identity, all of whose elements are bounded left Engel, is locally nilpotent.     

On finite groups with automorphisms whose fixed points are Engel

The main result of the paper is the following theorem. Let q be a prime, n a positive integer, and A an elementary abelian group of order q². Suppose that A acts coprimely on a finite group G and assume that for each \({a \in A^{\#}}\) every element of C_G(a) is n-Engel in G. Then the group G is k-Engel for some \({\{n,q\}}\)-bounded number k.     

On locally graded groups with an Engel condition on infinite subsets

A group G is said to be in E_k^*E_k^* (k a positive integer), if every infinite subset of G contains a pair of elements that generate a k-Engel group.¶It is shown that a finitely generated locally graded group G in E_k^*E_k^* is a finite-by- (k-Engel) group, in particular a finite extension of a k-Engel group.     

On sub‐Riemannian geodesics on the Engel groups: Hamilton's equations

We study the sub‐Riemannian geometry on the Engel group which is a step 3 nilpotent Lie group on . Our main result is to solve the Hamiltonian equations associated with the bi‐characteristic curves and express the solutions in terms of elliptic functions. Our model covers both the Heisenberg group and the Martinet case when setting certain parameters to be zero.     

The Engel Length of the Product of Engel Elements

Considering the relation between the Engel lengths of elements and their product, we give a counterexample to Question 11.88 in the “Kourovka Notebook” [1].