On commuting automorphisms of finite p-groups of coclass 3

Abstract

Let [Formula: see text] be a group. If the set [Formula: see text] forms a subgroup of [Formula: see text], then G is called [Formula: see text]-group. Recently, it has been proven that the minimum coclass of a non-[Formula: see text][Formula: see text]-group is equal to [Formula: see text]. In this paper, we deal with finite [Formula: see text]-groups of coclass [Formula: see text]. We prove that a finite [Formula: see text]-group [Formula: see text] of order [Formula: see text], [Formula: see text], of coclass [Formula: see text] for an odd prime [Formula: see text] is an [Formula: see text]-group except when the nilpotency class of the second term of upper central series [Formula: see text], [Formula: see text], is equal to [Formula: see text]. In particular, we show that in this case, center of [Formula: see text], [Formula: see text], and commutator subgroup [Formula: see text], [Formula: see text], are of order [Formula: see text] and [Formula: see text], respectively, and [Formula: see text] is a group of order [Formula: see text]. Also, we give some sufficient conditions on a finite 2-group [Formula: see text] of coclass [Formula: see text] implying that [Formula: see text] is an [Formula: see text]-group.