Some properties of certain automorphism groups of finite p-groups

Abstract

Let [Formula: see text] be a group and let [Formula: see text] denote the absolute center of [Formula: see text]. An automorphism [Formula: see text] of [Formula: see text] is called an absolute central automorphism if [Formula: see text] for each [Formula: see text]. An automorphism [Formula: see text] of [Formula: see text] is called a central automorphism if [Formula: see text] for each [Formula: see text]. Also, let [Formula: see text] be an autonilpotent finite [Formula: see text]-group of class [Formula: see text], where [Formula: see text]. We call an automorphism [Formula: see text] of [Formula: see text] is an [Formula: see text]th autoclass-preserving if for all [Formula: see text], there exists an element [Formula: see text] such that [Formula: see text], where [Formula: see text] is the [Formula: see text]th autocommutator subgroup of [Formula: see text]. In this paper, first, we characterize finite [Formula: see text]-group [Formula: see text] of class [Formula: see text] such that every central automorphism is absolute central. We also obtain a necessary and sufficient condition for an autonilpotent finite [Formula: see text]-group of class [Formula: see text] such that each absolute central automorphism is an [Formula: see text]th autoclass-preserving.