Nilpotent groups related to an automorphism

The aim of this paper is to state some results on an a-nilpotent group, which was recently introduced by Barzegar and Erfanian (Caspian J. Math. Sci. 4(2) (2015) 271-283), for any fixed automorphism a of a group G. We define an identity nilpotent group and classify all finitely generated identity nilpotent groups. Moreover, we prove a theorem on a generalization of the converse of the known Schur's theorem. In the last section of the paper, we study absolute normal subgroups of a finite group.