Equality of derival automorphisms and certain automorphisms in finitely generated groups

Abstract

Let [Formula: see text] be a group and [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] denote the group of all inner automorphisms, the group of all pointwise inner automorphisms, the group of all central automorphisms and the group of all derival automorphisms of [Formula: see text], respectively. We know that in a finite [Formula: see text]-group [Formula: see text] of class 2, [Formula: see text] if and only if [Formula: see text] is cyclic and [Formula: see text], where [Formula: see text] is the group of all derival automorphisms of [Formula: see text] which fix [Formula: see text] elementwise. In this paper, we characterize all finite nilpotent groups of class 2 for which [Formula: see text] or [Formula: see text] is equal to [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Also, we characterize all finitely generated nilpotent groups of class 2 for which [Formula: see text] is equal to [Formula: see text] and give some interesting corollaries in this regard.