Die $A$-Norm einer Gruppe

On Certain Weak Engel-Type Conditions in Groups

“…Now we restrict our attention to the n-Engel word w x y = x n y , where n is an integer greater than 1. Although the right 2-Engel elements of a group form a subgroup [7], Macdonald has shown in [9] that there exists a finite 2-group in which the set of right 3-Engel elements is not a subgroup. The set of right n-Engel elements of a group G coincides with the intersection of all the sets W w R g = a ∈ G a n g = 1 , with g ∈ G. Then, when n ≥ 3 the set W w R g cannot be a subgroup of G in general, but property ii holds in metabelian groups.…”

Criteria for the solubility of finite groups by their centralizers

“…By definition of N the centralizer C G (a) is normal in N for any element a ∈ G. Therefore a −1 a b ∈ C N (a) for any two elements a, b ∈ N . So N is a 2-Engel group (see also [3]). Now it is well known that N is a nilpotent group of class at most 3 and so it is soluble group of class at most 2.…”

On right n-Engel subgroups II

“…Let G be any group with a normal right 2-Engel subgroup H . In [8,9] (see also [10,Theorem 7.13 ∈ H , x, y, z ∈ G , is abelian and so [H, G, G, G] 2 = 1. Examples 1 and 2 in [6] then show that this is the best possible.…”