CI-Section And Solvability Of Maximal Subgroups Of Finite Groups

In 1998, in paper 《 on CI-section and C-index of finite groups 》 Shirong Li defined the concept of CI-section: Let G be a finite group and let M be a maximal subgroup of G,For each chief factor K/L of G such that L ≤ M and G = MK. we called the group M ∩ K/L a CI—section of M in G.All CI-sections of M are isomorphic.This paper contains three chapters .It centres on the influence of CI—section of maximal subgroup on the structure of finite group.In section one, the author defined the set of maximal subgroups of G: M_h(G)= {M < G | H (?) M}, here H is a normal subgroup of G,through discussing CI section of some special maximal subgroups of G with new ways, continue to do Shirong Li's work on CI-section , abtained some new results on solvability,p-solvability,π—solvability of H. what's more, generlized some relative results in paper 《 on Cl-section and C-index of finite groups 》 , for example , in Theorem2.2 of the same paper ,Shirong Li proved: A group G is solvable if and only if Sec(M) is nilpotent for every maximal subgroup M ∈ F_pc(G) where p is the largest prime divisor dividing the order of G. but in this thesis, as Theorem 2.3 shows: let G is a group, if for every maximal subgroup M G F_pc(G) ∩M_H(G),Sec(M) is nilpotent, where p is the largest prime divisor diving the order of G,then H is solvable. clearly the later theorem generalized the former .In section two, using new ideals,connected Deskins maximalcompletion with CI—section, abtained some new characterizations of solvability of a finite group G. for example, in paper 《 A note on the index complex of a maximal subgroup 》 Deskins proved that group G is solvable if and only if for every maximal subgroup M of G, there exists a maximal completion C such that C/K(C) is nilpotent with a Sylow2—subgroup of class at most 2. later, in the first paper Shirong Li proved :group G is solvable if and only if, for every maximal subgroup M of G,Sec(M) is 2-closed.but in theorem 3.6 of this dissertation, the author will prove: group G is solvable if and only if for every maximal subgroup M of G, either there exists a maximal completion C such that C/K(C) is nilpotent with a Sylow2-subgroup of class at most 2 or Sec(M) is 2-closed. one of the cases was showed in theorem 4.18: finite group G is solvable if and only if Sec(M) is nilpotent for every maximal subgroup At without prime index in G.In section 3, Through considering some special maximal subgroups of a givengroup G, using concept of CI—section ,the structure of G is discussed.for instance, in theorem 4.18: finite group G is solvable if and only if Sec(M) is nilpotent for every maximal subgroup M with neithor prime p nor p² index in G...