Gunnar Traustason

Powerful 2-Engel Groups

We study powerful 2-Engel groups. We show that every powerful 2-Engel group generated by three elements is nilpotent of class at most two. Surprisingly, the result does not hold when the number of generators is larger than three. In this article and its sequel, we classify powerful 2-Engel groups of class 3 that are minimal in the sense that every proper powerful section is nilpotent of class at most 2.

SYMPLECTIC ALTERNATING ALGEBRAS

This paper begins the development of a theory of what we will call symplectic alternating algebras. They have arisen in the study of 2-Engel groups but seem also to be of interest in their own right. The main part of the paper deals with the challenging classification of some algebras of this kind which arise in the context of 2-Engel groups and give some new information about these groups. The main result is that there are 31 such algebras with dimension 6 over the field of three elements.

Subnormality conditions in non-torsion groups

According to results of Heineken and Stadelmann, a non-torsion group is a 2-Baer group if and only if it is 2-Engel, and it has all subgroups 2-subnormal if and only if it is nilpotent of class 2. We extend some of these results to values of n greater than 2. Any non-torsion group which is an n-Baer group is an n-Engel group. The converse holds for n = 3, and for all n in case of metabelian groups. A nontorsion group without involutions having all subgroups 3-subnormal has nilpotency class 4, and this bound is sharp.

A note on conciseness of Engel words

It is still an open problem to determine whether the nth Engel word [x, n y] is concise, that is, if for every group G such that the set of values e n (G) taken by [x, n y] on G is finite it follows that the verbal subgroup E n (G) generated by e n (G) is also finite. We prove that if e n (G) is finite, then [E n (G), G] is finite, and either G/[E n (G), G] is locally nilpotent and E n (G) is finite, or G has a finitely generated section that is an infinite simple n-Engel group. It follows that [x, n y] is concise if n is at most four.

Groups that are Pairwise Nilpotent

In this article we study groups generated by a set X with the property that every two elements in X generate a nilpotent subgroup.

A Remark on the Structure of N-Engel Groups

We observe that there exists a positive integer c(n) such that for every locally nilpotent n-Engel group G we have that G/Z c(n)(G) is of n-bounded exponent. This strengthens a result of Burns and Medvedev.