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.