automorphism group, nth autoclass-preserving automorphism, absolute

Let g be a group and l(g) denotes the absolute center of g. ##an automorphism α of g is called an absolute central automorphism if x−1xα ∈ l(g), for each x ∈ g. ##let g be an autonilpotent finite p-group of class n + 1, where n ≥ 1. ##we call an automorphism α of g an nth autoclass- preserving if for all x ∈ g, there exists an element gx ∈ kn−1(g) such that xα = g−1 x xgx, where kn−1(g) is the n − 1th autocommutator subgroup of g. in this paper we obtain a necessary and sufficient condition for an autonilpotent finite p-group of class n + 1 such that each absolute central automorphism an nth autoclass-preserving.