The Normalizer Problem Of Integral Group Rings Of Finite Groups

The normalizer problem has become one of the most extensively studied top-ics in the theory of integral group rings of finite groups in recent years. A further motivation to study this problem came from the work of Hertweck, which linked the normalizer problem with the isomorphism problem for integral group rings. By using this intimate link between these two problems, Hertweck succeeded in constructing a counterexample to the isomorphism problem, which had lasted almost for sixties years sine it was first initiated by Higman in 1940. In this paper, based on other people's work, we carry out further investigations on the normalizer problem for integral group rings of finite groups.In Chapter 3, we investigate the normalizer property of special extensions—standard wreath products, of finite groups. It is shown that the normalizer prop-erty holds for wreath products of finite nilpotent by cyclic groups and for wreath products of finite nilpotent by some special 2-groups (PN 2-groups, dihedral and generalized quaternion 2-groups) as well. In Chapter 4, we investigate the nor-malizer property for general extensions of finite groups, such as extensions of nilpotent groups by groups whose integral groups rings having only trivial units, extensions of groups of odd order by groups whose integral groups rings having only trivial units, extensions of nilpotent groups by symmetric group of degree n, and extensions of groups of odd order by Symmetric group of degree n, etc. Our results obtained in this chapter generalize some ones of Li Yuanlin and a result due to Petit Lobao and Seghal.In the last three chapters, we devote to the study of some specific auto-morphisms of finite groups, which not only have intimate connections with the normalizer problem of integral group rings, but also are of interest in their own rights. In Chapter 5, we investigate class-preserving automorphisms of semidirect products of finite cyclic groups by maximal class 2-groups. Our results extend one result of Hertweck. In Chapter 6, we investigate the impact of the structure of finite groups, their special subgroups and their factor groups on Coleman auto-morphisms and obtain some sufficient conditions for OutCol(G) to be a p'-group. In Chapter 7, we investigate class-preserving Coleman automorphisms and C-automorphisms of finite groups. We prove that every C-automorphism of wreath products of nilpotent by abelian groups is inner, and so is every C-automorphism of wreath products of abelian by nilpotent groups. We also prove that every C-automorphism of semidirect products of finite abelian by maximal class p-groups is inner under a certain additional condition. Finally we prove that every class-preserving Coleman automorphism of some finite groups with special subgroups being TI-subgroups is inner.