θ~*-pairs And The Structure Of Finite Groups

The studies of the relationships between the structure of a finite group and its subgroups with certain properties is one of the most interesting topics in the theory of finite groups.Many scholars have made outstanding contribution to the field and a large amount of results have been obtained.Such achievements of the topics have effectively pushed forward the developments of group theory.The main purposes of this dissertation is try to research the structure of finite groups by using the concept ofθ-pairs andθ*-pairs for subgroups,which is motivated by the interesting concept of Index Complex introduced by W.E.Deskins.By the assumption of the properties ofθ-pairs,we obtain many interesting conclusions and generalize some known results.In this thesis,we first discuss the relationship between the structure of a finite group and the properties of the quotient group associated with theθ-pairs for some given subgroups.By considering 2-maximal subgroups together with solvability or normality of the quotient group related to theθ-pairs,we get some necessary and sufficient conditions for theπ-solvability,supersolvability or Sylow tower property of a finite group.Though one can characterize the structure of a finite group,we should also notice the fact that,in many cases,such results require some additional conditions imposed onθ-pairs.An obvious example is that the product of two subgroups associated withθ-pairs in a group is just the group itself.A nature question would be arose:whether or not such condition could be removed? For the purpose,we introduce the concept ofθ*-pairs for any subgroup of a group.Using the concept,we can make an efficient research of the relationship between the structure of a group and the properties ofθ*-pairs for the maximal subgroups of Sylow subgroups of some subgroups.By investigating the properties of theθ*-pairs for such subgroups,we further obtain some necessary and sufficient conditions for a group to be solvable,p-nilpotent or nilpotent,and some results in this section are extended to formations.