New LA-Groups And Automorphism Groups Of Meta-Cyclic P-Groups

Let p be odd prime. This thesis mainly studies some problems about the automorphism groups of finite p -groups. Based on the work of the formers, this thesis determines the structures of groups, calculates the orders of automorphism groups and deduces their constructions by applying free group theory, generation algorithm of p -groups, Schreier's extension theory of group, defining relation, the influence of some characterizations of subgroups on the structures of finite groups, number theory and combinatorial mathematics method. The thesis includes the following three chapters:Chapter one mainly analyses the research background and the present research situation of the automorphism groups of finite groups (especially p -groups).Chapter two first obtains two important infinite classes of finite p -groups by using Schreier's extension theory of group, then gives some important properties of them and further proves they are all LA-groups with homological method, thereby gets new series of LA-groups. Then this chapter discusses the relations between parameters in the defining relations and calculates the orders of the corresponding automorphism groups, this not only establishes LA-group properties of these two classes of groups, but also extends some known conclusions in recent years about the orders of automorphism groups. Chapter Three mainly investigates the automorphism groups of meta-cyclic p -groups. In process of this study, some new method are obtained which such that not only some new results of group theory are found, but also a few conclusions of combinatorial mathematics are got in the thesis. In detail, on the one hand, this chapter gives the orders and the structures of the automorphism groups of all classes of meta-cyclic p-groups which have been obtained by M. F. Newman and M. Y. Xu in 1987. On the other hand, as applications of the properties of meta-cyclic p -groups, some new significant combinatorial formulas are obtained from the new calculating techniques. Thus the techniques which used in this chapter also have important value for combinatorial mathematics.