The Influence Of Self-centralizing Subgroups On The Structure Of Finite Groups

In the investigation of finite groups,using properties of subgroups to protray the struc-ture and discuss properties of finite groups is a main direction and a common approach.In this thesis,we explored the properties of G by using properties of self-centralizing subgroups of G,and we got some new conclusions of SCT-groups and SCS-groups.According to the contents,this paper was divided into three chapters.In the first chapter,we mainly gave out concepts of the SCT-groups and SCS-groups and introduced their research background and some research results of predecessors.In the second chapter,we used the self-centralizing subgroups to research the structure of the SCT-groups and SCS-groups.If G is an SCT-group then the subgroups and the factor groups of G are also SCT-group:Theorem 2.1.1 Let G be a finite group.If G is an SCT-group,H?G,then H is also an SCT-group.Theorem 2.1.2 Let G be a finite group,N<G.If G is an SCT-group,then G/N is also an SCT-group.We also got that an SCT-group is nilpotent or is an F-group and orther new results.Theorem 2.1.6 Assume that G is an SCT-group.Then one of the statements holds.(1)G is nilpotent.(2)G=NH is a Frobenius group with a kernel N and a complement H.and N is the unique minimal normal subgroup of G and H is nilpotent.In particular,G is a solvable CN-group.Theorem 2.1.7 Let G be a nilpotent group.Then all self-centralizing subgroups of G are TI-subgroups if and only if cl(G)?2.As for SCS-groups,we got the following two new results.Theorem 2.2.1 Let G be a finite group,N<G.If G is an SCS-group,then G/N is also an SCS-group.Theorem 2.2.2 Assume that G is an SCS-group.Then G is a supersolvable group.When we studied the structure of SCS-groups,we get two new sufficient conditions of supersolvable groups.Here are the conclusionsTheorem 2.2.3 Let G be a finite group whose non-abelian and non-normal maximal subgroups are conjugated.Then G is solvable.Theorem 2.2.4 Let G be a finite group.If the number of the conjugacy class of non-normal subgroups is less than the number of the conjugacy class of maximal subgroups7,then G is solvable.The third part was mainly about a generalization of a theorem of Gagola and Lewis.In[24]Gagola and Lewis proved thst a finite group G is nilpotent if and only if X(1)2 di-vides |G:KerX| for all irreducible characters X%of G.In this part7,we prove the following generalization that a finite group G is nilpotent if and only if X(l)2 divides |G:KerX| for all monolithic characters X of G.Theorem 3.1 A finite group G is nilpotent if and only if x(1)2 divides |G:KerX| for all X?Irrm(G)).