Integral Group Rings Of Finite Groups And The Normalizer Problem

The normalizer problem is a famous problem in the field of group rings.The problem is very important and closely related to another famous problem in group rings,namely isomorphism problem.Therefore it is a significant research topic to find the classes of finite groups such that they have the normalizer property.In chapter ? we introduce the progress of the normalizer problem in recent years and in chapter ? we give some preliminary knowledge for the study of this problem,preparing for the following three chapters.In chapter ?,we first prove that Outc(G)of an AZ-group G must be a 2'-group and therefore the normalizer property holds for AZ-groups.Then we find some classes of finite groups such that the intersection of their outer class-preserving automorphism groups and outer Coleman automorphism groups is 2'-groups and therefore the normalizer property holds for these kinds of finite groups.Finally we show that the normalizer property holds for the wreath products of AZ-groups by rational permutation groups under some conditions.In Chapter ?,we prove that if G is a finite solvable group with semidihe-dral Sylow 2-subgroups,then Outc(G)? OutCol(G)is a 2'-group and therefore G satisfies the normalizer property.As some applications of this result,we also investigate the normalizer property of the following groups:the groups whose Sy-low 2-subgroups are semidihedral and Sylow subgroups of odd order are all cyclic,the groups G=N × S with N a nilpotent normal subgroup and S a maximal class 2-group,and the wreath products G=H(?)Q with H a group whose Sylow 2-subgroups is of maximal class with order?8 and Q a rational permutation group.Recall that a subgroup H of a finite group G is said to be a weak second maximal subgroup of G if there exists a maximal subgroup M of G such that H is a maximal subgroup of M.Let m(G,H)denote the number of maximal subgroups of G containing H.In the last Chapter we give the classification of G with G/CoreG(H)solvability and H a weak second maximal subgroup of G when m(G,H)-1 is equal to the index of some maximal subgroup of G,which lead to proving the normalizer property of G/CoreG(H)and make us understand the difference between the weak second maximal subgroups and the second maximal subgroups.