Influences Of The Cover-Avoidance Properties On The Structure Of Finite Groups

It is important to study the structure of finite groups by maximal subgroups and maximal subgroups of Sylow subgroups with some properties in the researches on finite groups.Many scholars have studied these respects and given many important results,for example,the famous Huppert's theorem,i.e.,a finite group G is supersolvable if and only if every maximal subgroup of G has prime index;a finite group G is nilpotent if and only if every maximal subgroup of G is normal in G.Srinivasan[1]proved a typical result:a finite group G is supersolvable provided that every maximal subgroup of Sylow subgroup is normal in G,and so on.These results have been widely generalized.The research was paid more attention and gained lots of famous results by reducing the number of maximal subgroups(maximal subgroups of Sylow subgroups)and changing the normality of theirs.In this paper,we study the properties of normality,CAP-subgroup and SCAP-subgroup in finite group,with which we describe the structure of some finite groups by studying the maximal subgroups and maximal subgroups of Sylow subgroups,and get some meaningful results.We divide this paper into four chapters.The main of the paper is the following:In Chapter One,We introduce the backgrounds and the present investigations of this paper.In Chapter Two,With".M_d(P)" we study the structure of finite groups by CAP-subgroups and SCAP-subgroups.We obtain some sufficient or sufficient and necessary conditions on(p-)supersolvability,p-nilpotency.And these results largely generalize some known results.In Chapter Three,We mainly study SCAP-property of some maximal subgroups, and get some necessary and sufficient conditions on solvability of finite groups.In Chapter Four,We define and study the generalized concept of CAP-subgroup: CAP~*-subgroup,with the concept we study the influence of some subgroups on the structure of finite groups.In fact,some interesting results have been made.