Coleman Automorphism Of Finite Group

Coleman automorphism occured first in the study of the normalizer)ZG(U of G in the units)ZG(U of ZG. Let G be a finite group and ? be an automorphism of G, ?is said to be a Coleman automorphism if the restriction of ? to any Sylow subgroup of G equals the restriction of some inner automorphism of G. Denote by(G)AutColthe group formed by all Coleman automorphism of G. It is clear that)G(Inn is a normal subgroup of)G(AutCol. We set)G(Inn/)G(Aut)G(OutColCol?.Dade E C. made enormous researches on Coleman automorphism of a finite group and obtained a lot of initial results, especially proved that Coleman outer automorphism group is nilpotent. Recently, Hertweck M. and Kimmerle W. proved that Coleman outer automorphism group is abelian and gave some sufficient conditions on the group being a’p- group and trivial group.Based on Coleman’s results, we obtained that the Coleman automorphism of finite nilpotent groups is trivial group. What about the Coleman automorphism of holomorphs of finite nilpotent groups? In this article, it proved that the outer Coleman automorphism of holomorphs of finite nilpotent groups is trivial group, i.e. every Coleman automorphism of holomorphs of finite nilpotent groups is an inner automorphism; The outer Coleman automorphism of inner nilpotent groups is trivial group.