The Investigation Of Prime Power Order Subgroups And The Structure Of Finite Groups

The relationship between the subgroups of a finite group G and the group G itself has been extensively studied in the literature. It is well known that the prime power order subgroups play a crucial role in the investigation of finite group theory. Our investigation focuses on the influence of minimal subgroups of Fitting subgroup and generalized Fitting subgroup, some maximal subgroups of Sylow subgroups on the structure of finite groups.The whole thesis, divided into three chapters, contains two parts. For detail, as the following:In Chapter One, we introduce the backgrounds and the present investigations of this paper.In Chapter Two, we recall that Miao L. and Guo W. firstly introduced the concept of F-s -supplement in " Finite groups with some primary subgroups F-s -supplemented" (Comm.in Algebra, 2005, Vol.33, No.8, 2789-2800).Definition l[l] Let F be a class of groups. A subgroup H of G is called F-s-supplemented in G, if there exists a subgroup K of G such that G = HK and K/(K∩ H_G) ∈ F. In this case, K is called an F-s-supplement of H in G.Particularly, we call that H is supersolvable-s-supplemented in G, if there exists a subgroup K of G such that G = HK and K/(K∩ H_G) is supersolvable. Similarly, H is p-nilpotent-s-supplemented in G, if there exists a subgroup K of G such that G = HK and K/{K∩H_G) is p-nilpotent for some prime p.we can see that supersolvable-s-supplemented subgroup is distinct to complemented subgroup.One side, supersolvable-s-supplemented subgroup is not complemented subgroup.Exemple : Let G be a cyclic group of p~n order and n > 1. We know that G is supersolvable, G has unique minimal subgroup X and X is supersolvable-s-supplemented in G. But X is non-complemented in G.