Some Sufficient Conditions For A Finite Group To Be Solvable

In this paper we study the influence of properties of subgroups on the structure of finite groups.In the first chapter,we introduce the research backgrounds.In the second chapter,we introduce some basic conceptions and lemmas which are needed.In the third chapter,the influence of K(G)on finite groups has been studied and some new results are obtained.In the four chapter,a character theoretic condition characterizing finite Frobenius groups is obtained by using the relationship between the actions of finite groups on its irreducible characters and conjugacy classes.The results are as follows:Theorem 3.1 Let G be a solvable group with K(G)= {1,m,m+2},and H,N be the normal subgroups of G with ?(H)=m,?(N)=m+2.Assume that G is abelian.Then one of the following holds:(1)If H ?N= 1,then G=H × N,|H|=p,|N|=q,and q-q=2;(2)If H ?N?1,then K(G)= {1,2,4},and G(?)Z4 × Z2,Z2 × Z2 × Z2,or G(?)Z8.Theorem 3.2 Let G be a nonabelian p-group with K(G)= {1,m,m+2}.Then G has order 33,and one of the following holds:(1)G=<a,b | a32 =b3=1,b-1 ab=a4);(2)G=<a,b | a3= =c3=1,[a,b]=c,[a,c]=[b,c]=1>.Theorem 3.3 Let G be a nonabelian non-p-solvable group.Then K(G)= {1,m,m+2}if and only if one of the following holds:(1)G is Frobenius group,G' is the kernel and the minimal normal subgroup of G,G'has cyclic complement of odder 9 or pq in G,(p,q)= 1,q=(p-1)/2;(2)G/N(?)Zp ×C3n.N is the unique minimal normal subgroup of G,and N is p-group,and for every x?G-N,|CG(x)|=9.Theorem 3.4 Let G be an insolvable group.Then K(G)= {1,m,m+2} if and only of one of the following holds:(1)|G/G'|= 9,,G' is insolvable and G' is the minimal normal subgroup of G,and|CG(x)|=9 for every element x?G-G' with order 3.(2)G=G' × Z(G),|Z(G)|is prime,G' is a simple group and ?(G')=m+2;(3)G' and G" are two normal subgroups of G,G" is insolvable and G/G"(?)Zp×C3n,p=(3n-1)/2,p is prime,and for every x?G'-G",|CG(x)|= 1 or |CG(x)|=3.Theorem 3.5 Let G be a nilpotent group.Then |K(G)|=4 if and only if G has order p4,and ?(M)=?(N)for every non-trivial normal subgroups M,N of G with |M|= |N|.Moreover,K(G)= {1,p,p2,p3},K(G)= {1,p,2p-1,p2+2p-2}or K(G)= {1,p,p2,2p2-1}.Theorem 3.6 Let G be a non-nilpotent solvable group,G' is the unique minimal normal subgroup of G.Then |K(G)|=4 if and only if G is a Frobenius group of type Cpn×Irr Cqr or Cpn ×Irr Cq3.Theorem 3.7 Let G be a non-nilpotent soluble group,G at least has two minimal normal subgroups,and G' is not the minimal normal subgroup of G.If ?(M)=?(N)holds for any two minimal normal subgroups M,N of G,then |K(G)|= 4 if and only if G is of type(Cpm × Cqn)×Irr and M,N,G' are all the nontrivial normal subgroups of G.Moreover,K(G)= {1,1+(pm-1)/R,1+(qn-1)/r,1+(pmqn-1)/r}.Theorem 4.2 Let G be a solvable group.If every ?? Irrm(G)is quasi-primitive,then G is abelian.Theorem 4.3 Let G be an M-group,and l=dl(G)the derived length of G.Then G is a relative M-group with respect to G(l-1).