Generalized Cover-avoidance Subgroups And The Structure Of Finite Groups

The relationships between the structure of a finite group and its subgroups having certain properties of normality have been extensively studied by many authors since the last two decades.Not only new concepts have been introduced but fruitful results have also been obtained.The achievements of this topic in normality have indeed pushed forward the developments of group theory.Among the modified concepts of normality,the concepts of c-normality and cover-avoiding property have attracted much attentions of many authors,however,we aware that these two important concepts are not explicitly related and linked.As a generalization of these concepts,Fan,Guo and Shum have first considered the semi cover-avoiding property of a group which generalizes both the cover-avoiding property and c-normality property,and subsequently,they obtained some interesting results.In Chapter 2 of this thesis,we concentrate on the p-nilpotency, p-supersolvablity and supersolvablity of a finite group G by assuming that the maximal and minimal subgroups of some Sylow subgroup of G having the semi cover-avoiding property.As a consequence,some known results in the literature are generalized.Moreover,some of them are extended to formations.In Chapter 3,we generalize the cover-avoiding subgroups in another direction and we call them the CAP-embedded subgroups.It is clear that cover-avoiding subgroups and permutably embedded subgroups are both CAP-embedded subgroups but not conversely.On the other hand,we note that the CAP-embedded subgroups are not necessarily semi cover-avoiding subgroups and vice versa.In this Chapter,we characterize the structure of finite groups by considering some of their subgroups having the CAP-embedded property.Traditionally,the research of finite groups are devoted on the supersolvablity and p-nilpotentcy of a given G which are based on the maximal and minimal subgroups of the Sylow subgroups of G having some generalized normality property.Recently,Skiba assumed that every Sylow subgroup P of G has a subgroup D such that 1＜|D|＜|P| and all subgroups H of P with order |H|=|D| and with order 2|D|(if P is a nonabelian 2-group and |P:D|＞2) have some subgroups property in G.He then described the structure of the finite group G under the above assumptions.In particular,some sufficient conditions are given for G to be supersolvable.In Chapter 4,we further obtain some conditions for a finite group G to be p-nilpotent, p-supersolvable or supersolvable by considering the same subgroups having the normally embedded property in G.It is noteworthy that the subnormal subgroups is one of the most important subgroups in a finite group G.This kind of subgroups has many nice properties and they are needed to determine whether a finite group is nilpotent or not? In this aspect,Xu and Zhang have obtained some results by using the class of conjugate-permutable subgroups.Since conjugatepernmtable subgroups must be subnormal,the influence of subnormal subgroups in the structure of finite groups are described in Chapter 6.Some necessary and sufficient conditions for a finite group to be nilpotent are given.We note that all of the obtained results can be further generalized to formations.The structure of a finite group G whose subgroups are normal were first described by Dedekind and Bare.The finite groups whose subgroups are quasinormal were later determined by Georges.It is clear that every subgroup of a finite group G is s-quasinormal if and only if G is nilpotent.The structure of finite solvable groups to be T-groups,PT-groups and PST-groups were studied by Gaschiitz,Zaeher and Agrawal,respectively.In Chapter 5,the structure of finite solvable groups whose subgroups having transitivity property on weakly c-normal subgroups,weakly s-pernmtable subgroups,c-supplemented subgroups, weakly s-supplemented subgroups,respectively are further studied.Finally,the structure of a finite group G whose subgroups having weakly c-normal property, weakly s-permutable property,c-supplemented property,weakly s-supplemented property,respectively are described.