nilpotency and solubility of groups relative to an automorphism

in this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. with this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. n(g) and s(g) are the set of all nilpotency classes and the set of all solvability classes for the group g with respect to different automorphisms of the group, respectively. if g is nilpotent or solvable with respect to the all its automorphisms, then is referred as it absolute nilpotent or solvable group. subsequently, n(g) and s(g) are obtained for certain groups. this work is a study of the nilpotency and solvability of the group g from the point of view of the automorphism which the nilpotent and solvable groups have been divided to smaller classes of the nilpotency and the solvability with respect to its automorphisms.