Automorphism On The Class Of Finite Group Remains Some Results

The study of class-preserving automorphisms has been one of the hottest subjects in the theory of finite groups. Based on the existing results in the literature, further research on class-preserving automorphisms is carried on in this paper and related results on several aspects are obtained which we state as follows:In chapter 2, the properties of the direct product decompositions of class-preserving automorphism groups are presented and it is deduced that Outc(G)=1 for a finite group G if and only if the class-preserving outer automorphism group of each direct product factor ofG equals 1, which plays an important role in the study of the normalizer property. In chaper 3, the relations between class-preserving automorphism groups and central automorphism groups of finite groups are established, and by means of the properties of central automorphisms it is shown that ifG is a finite p-group of nilpotent class 2 and its derived subgroup is a cyclic group, then the class-preserving outer automorphism group of G equals 1,and also the class-preserving outer automorphism groups of finite non-abelian p-groups satisfying |G/Z(G)|≤p4 are investigated.In chapter 4, the class-preserving automorphisms of finite groups which are extensions of abelian groups by dihedral groups are been studied. In the final chapter, it is shown that the class-preserving outer automorphism group of each finite p-group G with |G|≤p4 is trivial.