Commuting automorphisms of finite 2-groups of almost maximal class, II

Let [Formula: see text] be a finite [Formula: see text]-group. In our recent paper, it was shown that in a finite [Formula: see text]-group of almost maximal class, the set of all commuting automorphisms, [Formula: see text] is a subgroup of [Formula: see text]. Also, we proved that the minimum coclass of a non-[Formula: see text], [Formula: see text]-group is equal to 3. Using these results, in this paper, we will take of the task of determining when the group of all commuting automorphisms of all finite [Formula: see text]-groups of almost maximal class are equal to the group of all central automorphisms. This determination is not easy. We will prove they are equal, except only for five ones. We show that the minimum order of a [Formula: see text]-group which it’s group of all commuting automorphisms is not equal to it’s group of all central automorphisms is [Formula: see text]. Also, we prove that if [Formula: see text] is a finite [Formula: see text]-group in which [Formula: see text], then the subgroup of right 2-Engel elements of [Formula: see text], [Formula: see text], coincides with the second term of upper central series of [Formula: see text].

Engel groups with an identity

We give an affirmative answer to the question whether a residually finite Engel group satisfying an identity is locally nilpotent. More generally, for a residually finite group [Formula: see text] with an identity, we prove that the set of right Engel elements of [Formula: see text] is contained in the Hirsch–Plotkin radical of [Formula: see text]. Given an arbitrary word [Formula: see text], we also show that the class of all groups [Formula: see text] in which the [Formula: see text]-values are right [Formula: see text]-Engel and [Formula: see text] is locally nilpotent is a variety.

Linear Groups with Almost Right Engel Elements

Let G be a linear group such that for every g ∈ G there is a finite set ${\cal R}(g)$ with the property that for every x ∈ G all sufficiently long commutators [g, x, x, …, x] belong to ${\cal R}(g)$. We prove that G is finite-by-hypercentral.