Subnormality conditions in non-torsion groups

Abstract

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 the case of metabelian groups. A non-torsion group without involutions having all subgroups 3-subnormal has nilpotency class 4, and this bound is sharp.

Locally nilpotent 4-Engel groups are Fitting groups

“…According to a result of H. Heineken [9], a non-torsion group is a 2-Baer group if and only if it is 2-Engel. In [12] it is shown that the analogous result holds for the integer 3. Thus a non-torsion group is 3-Engel if and only if it is a 3-Baer group.…”

Centre-by-metabelian groups with a condition on infinite subsets

“…Denote by B * k the class of groups such that every infinite subset contains an element x such that x is subnormal of defect k. It is proved in [8,Corollary 2.5] that a metabelian non-torsion group is a k-Baer group (that is every cyclic subgroup of G is subnormal of defect k) if, and only if, G is a k-Engel group. Here, using Theorem 1.2, we shall improve this result with the following: Proof: Let G be a finitely generated nilpotent group of class at most k+2 and suppose that G is in E * k+1 .…”

“…We have [x, k+1 y] ∈ H and For the proof of Theorem 1.3, we need further lemmas. Note that it is proved in [8,Theorem 2.3] that every non-torsion k-Baer group is a k-Engel group. But the converse is shown only in the metabelian case.…”

On 3-Baer groups

“…From (5) we have that H i+2 = {1} and thus (6) gives us (ty) 3 i+2 = t 3 i+2 y 3 i+2 as we wanted. P It is well known [3] that metabelian 3-Engel groups without involutions are nilpotent of class at most 3.…”

“…If G is a non-torsion group, that is a group that contains an element of infinite order, then we have the stronger result that G is a 3-Engel group [5]. Theorem 1 is thus essentially a theorem about 3-Baer p-groups.…”