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

Abstract

Let [Formula: see text] be a group. If the set [Formula: see text] for all [Formula: see text] forms a subgroup of [Formula: see text], then [Formula: see text] is called [Formula: see text]-group. Let [Formula: see text] be an odd prime. Recently it has been proven that a finite [Formula: see text]-group of almost maximal class is an [Formula: see text]-group. For finite 2-groups of almost maximal class, the situation is much more complicated. This paper deals with the case [Formula: see text]. We prove that a 2-group of almost maximal class is an [Formula: see text]-group. We show the minimum coclass of a non-[Formula: see text], [Formula: see text]-group is equal to 3. We also discuss the direct product of certain [Formula: see text]-groups and subsequently we give some applications of our results.