Unipotent automorphisms of solvable groups

1 Introduction

If G is a group and a is an automorphism of G, we say that a is a nil-automorphism if, for every g∈G, there exists n=n⁢(g) such that [g,na]=1 (here commutators are taken in the holomorph G⁢Aut⁢(G) and, as usual, the n-fold commutator [g,na] is defined recursively by [g,0a]=g and, for n>0, [g,na]=[[g,n-1a],a]). When n can be chosen independently from g, we say that a is n-unipotent or simply unipotent if the integer n is clear. The notion of nil and unipotent automorphism was defined by Plotkin in [9] and, in recent years, some papers have dealt with this topic.

Nil and unipotent automorphisms can be regarded as a natural extension of the concept of Engel element, since a nil-automorphism a is just a left-Engel element in G⁢〈a〉. Another way to look at nil-automorphisms, is to consider them as a generalization of unipotent automorphisms of vector spaces. For these reasons there are several natural questions that can be asked about nil-automorphisms, which are suggested by known facts about Engel groups or unipotent linear groups. Maybe the first question we may ask is whether a group of automorphisms whose elements are n-unipotent, needs to be locally nilpotent. This question is just the automorphism version of the well-known problem about the local nilpotency of n-Engel groups. This problem is still open, although some interesting partial results have been obtained either for small values of n or for groups in some particular classes. When n is smaller than 4, the problem has positive solution. The cases n=3 was treated by Heineken in [7] . The case n=4 is considerably more involved and was solved in [6].

Engel groups in some special classes have been considered by several authors. In [1] Baer studied the subset of left Engel elements in groups satisfying Max, proving that this subset is actually the Fitting subgroup. The set of Engel elements (right and left) in solvable groups was later investigated by Gruenberg in [5]. Another remarkable result was obtained in [10] where the authors show that profinite Engel groups are locally nilpotent. This was later extended to compact Hausdorff groups by Medvedev in [8].

Recent results on unipotent action are contained in [2] and[4]. The first paper is concerned with groups of unipotent automorphisms of finitely generated residually finite and profinite groups. In this situation the authors prove that the group of automorphisms is locally nilpotent. Frati looks at automorphisms of solvable groups. When the group acted upon is abelian-by-polycyclic or has finite Prüfer rank, then Frati shows that a group of n-unipotent automorphisms is locally nilpotent. Both papers [2] and [4] rely heavily on results by Crosby and Traustason on normal right-Engel subgroups (see [3]). In turn, in their work, Crosby and Traustason use some deep results on Lie algebras due to Zelmanov.

In this paper we continue the study of unipotent automorphisms of solvable groups proving the following theorem.

2 The results

We start with two definitions.

For a group G, subgroup H of G and a non-negative integer n, we define recursively [H,nG] by [H,0G]=H and [H,nG]=[[H,n-1G],G] for n>0.

We are now in a position to prove our main result.

It is worth pointing out that, in general, the fact that a group of automorphisms acts n-unipotently, is not enough to ensure that it stabilizes a finite series.

Clearly in the example we are going to discuss, the group H will not be finitely generated and G will not have any series with torsion-free factors (indeed G is going to be a torsion group). Let H be a countable elementary abelian 2-group and A a cyclic group of order 2. Let G=wr⁡AH the standard wreath product of A and H. The group H acts on G as a group of 3-unipotent automorphisms but it does not stabilize any finite series.

Let m be the smallest integer such that the Burnside variety ℬ⁢(2m) of groups of exponent dividing 2m is not locally finite. Choose any finitely generated infinite group G in ℬ⁢(2m) and pick an involution a∈G. For every g∈G the subgroup D=〈a,ag〉 is a dihedral group and x=ag⁢a=[g,a] generates a normal subgroup of D. Moreover, xa=x-1 and a straightforward calculation shows that, for every k, one has

For k=m+1 we obtain

This means that a is an n-unipotent automorphism of G for n=m+1. We claim that there exists at least one involution a that does not stabilize any finite series in G. By way of contradiction assume that this is not true. If u is any element of G, then a=u2n-1 is either 1 or an involution. If a is not trivial, then it stabilizes a finite series

Therefore we have a decreasing chain

from which we get the subnormal series

Since every subnormal cyclic group is contained in H⁢(G), the Hirsch–Plotkin radical of G, we have shown that

Therefore G/H⁢(G) has exponent dividing 2n-1 and, by our choice of n, it is finite. Hence H⁢(G) is finitely generated whence nilpotent and finite, being a torsion group. This, in turn, implies that G is finite, a contradiction.