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

Abstract

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].