Research On Some Questions In The Theory Of Groups

The main object of this paper is to attack some open questions in the theory of groups.In Chapâ… ,we introduce the background of the paper and some main results obtained in this paper.In Chapâ…¡,we introduce the notations and terminologies used in this paper and we list also some known results.The main purpose in Chapâ…¢is to research covering subgroups and injectors. In this chapter,we solve an open question on(?)-covering subgroups proposed by L. A.Shemetkov and an open question on(?)-injectors proposed by Wenbin Guo.The results in this chapter also shed new light on a problem of Unsolved Problems in Group Theory-THE KOUROVKA NOTEBOOK(Problem 12.96).We study the modular law on Fitting class in Chapâ…£.In Unsolved Problems in Group Theory-THE KOUROVKA NOTEBOOK,there is such an open problem (Problem 14.47):"Is the lattice of all soluble Fitting classes of finite groups modular?" Work on this Problem,in this chapter,we proved that:Let(?)=LR(x),(?)=LR(y)and(?)=LR(f)be local Fitting classes with least H-function x,y and f respectively,and x≤f.If x and y satisfy that x(p)∨y(p)= S_n(G|G=G_x(p)G_y(p)for all p∈P where x(p)≠φand y(p)≠φ,then the following modular law holds:((?)∨_l(?))∩(?)=(?)∨_l((?)∩(?)).In Chapâ…¤,we research a Shemetkov's question on Fitting class.In the universe (?)_Ï€(?)_π＇—the class of allÏ€-soluble group,we give a positive answer to this question. We prove that:For any set of primesÏ€- and any local Fitting class(?),the Fitting class K_Ï€((?))is a local Fitting class.By using this result,we give some applications.In particular,the(?)-radical of a HallÏ€-subgroup of a finite soluble group is described.In Chapâ…¥,we give some new characterizations of finite supersoluble groups by using the property of X-semipermutable subgroups.In particularly,we prove that if all 2-maximal subgroups of a group G is F(G)-semipermutable in G,then G is supersoluble. This theorem give a positive answer to an open question proposed by Wenbin Guo,A.N.Skiba and K.P.Shum on Journal of Algebra.Under the condition that all Sylow subgroups of G are F(G)-semipermutable in G,we obtain that G is supersoluble and,furthermore,the factor G/O_p＇,p(G)is cyclic for any prime p.Also,we prove that a group G is supersoluble if all maximal subgroups of its Sylow subgroups are F(G)-semipermutable in G.In this chapter,by using the F(G)-semipermutability of 2-maximal subgroups of some Sylow subgroups,we obtain a criterion on p-nilpotent groups.In Chapâ…¦,we give a definition on s-conditionally permutable subgroups.By this notation,we give some criterions of groups belonging to some given saturated formation. We also give some characterizations on p-supersoluble groups.For example, we prove:if G is a p-soluble group,then G is p-supersoluble if and only if one of the following statement hold:1)for any non-Frattini p-chief factor H/K of G,there exists a maximal subgroup P₁ of a Sylow p-subgroup of G such that P₁ is s-conditionally permutable in G and does not cover H/K.2)there exists a normal subgroup N of G such that G/N is p-supersoluble and any maximal subgroup of a Sylow p-subgroup of N either has a p-supersoluble supplement in G or is s-conditionally permutable in G.