Research On Some Problems Of Finite Groups And Finite Dimensional Lie Algebras

The contents of investigation of this dissertation mainly include two aspects.First of all,we study the structure of finite groups by means of the conjugacy classes of non-cyclic subgroups and non-abelian subgroups,the automizers of a-belian subgroups and the global breadth in the sense of Frobenius and get many meaningful new results which extend some known results.Secondly,there is a tradition of investigating analogies between group theory and the theory of Lie algebras.There are myriad correspondences between these two fields.Following the analogy with group theory,we investigate the properties of finite dimensional Lie algebras in terms of the normalizer of nilpotent residual.In chapter 2,let π(G)denote the set of the prime divisors of |G| and δ(G)the number of conjugacy classes of all non-cyclic subgroups of G.Firstly,we give complete classification of finite groups G such that δ(G)=2|π(G)1-2 andδ(G)=2|π(G)|-1.Secondly,we give a complete classification of finite solvable groups with δ(G)=|π(G)|+2.Finally,it was showed that δ(G)≥M(G)+2 for unsolvable groups G and the equality holds if and only if G≌A5 or SL(2,5),where M(G)denotes the number of conjugacy classes of all maximal subgroups of G.In chapter 3,we investigate the relations on conjugacy classes of non-abelian subgroups and the solvability of finite groups.Firstly,we prove that if G has at most two conjugacy classes of non-abelian maximal subgroups,then G is solvable.In addition,let τ(G)denote the number of conjugacy classes of all non-abelian subgroups of G and Γ(G)the number of conjugacy classes of all non-normal and non-abelian subgroups of G.We show that the finite groups G with τ(G)≤M(G)or Γ(G)≤|π(G)|-1 are solvable.Finally,we give the lower bounds of the number of conjugacy classes of all non-abelian subgroups of non-prime-power order of G in terms of π(G).In chapter 4,let H be subgroup of G.The automizer of H in G is AutG(H):=NG(H)/CG(H).AutG(H)can be regarded as a subgroup of Aut(H)and AutG(H)contains an isomorphic copy of Inn(H).A group G is called an ANC-group if AutG(A)≌Inn(A)or AutG(A)≌Aut(A)for every abelian subgroup A of G.In this chapter,we characterize the nilpotent ANC-group and ANC-group with abelian Sylow subgroup.In chapter 5,let Div(G)denote the set of all divisors of |G|.For e∈Div(G),set Le(G)={x∈G|xe=1}.Frobenius showed |Le(G)|=kee for some positive integer ko.B(G)=max{|Le(G)|/e|e∈Div(G)} is called global breadth in the sense of Frobenius.In this chapter,we describe the structures of groups with B(G)=4 and show that the groups with B(G)<7 are always solvable.In particular,the non-abelian simple groups with B(G)=8 must be isomorphic to A5.In chapter 6,let L be a finite dimensional Lie algebra over field IF with char-acteristic 0 and L∞ denote the nilpotent residual of L.In the first part,we study the property of L∞ and prove that L is nilpotent if and only if L∞(?)NL(M)for each maximal subalgebra M of L.In the second part,we introduce the notion of F_n-Lie algebra,that is,L is said to be an F_n-Lie algebra if L∞ is nilpotent.We define S(L)=(?)NL(H∞).Furthermore,let S0(L)=0 and H<L Si+1(L)/Si(L)=S(L/Si(L))for i≥1,then {Si(L)} is an ascending series of ideal of L.Set S∞(L)=(?)Si(L).It was proved that L is an F_n-Lie algebra if and only if L=S∞(L).Finally,L is called an S-Lie algebra if L=S(L).The basic properties of S-Lie algebras are characterized and the conditions of sufficient on S-Lie algebra are gained.