| It has been one of the important topics to study the structure of the finite groups using various properties of subgroups for a long time.The normality of subgroups is a basic property in finite group theory.This leads to a lot of generalized normalities and a large number of research significant results are obtained.Recently,it is a hot topic to study the properties of finite groups by using the properties of some subgroups.In this paper,we continue to study generalized normalities and investigate the structure of finite groups by using the self-centralizing subgroups.Many interesting results have been obtained,and they generalize some known and important conclusions.This paper consists of four chapters.In the first chapter,we introduce the research background.In the second chapter,we introduce some basic concepts and lemmas.The third chapter is divided into three parts:In section 3.1,we investigate the finite groups whose self-centralizing subgroups are all C-normal subgroups.Some sufficient conditions and related structures of solvable and supersolvable groups are obtained.Definition 3.1.1 Let G be a finite group.If every self-centralizing subgroup of G is C-normal in G,then we call G is an SCCN-group.CN-groups and SCN-groups are SCCN-groups.Theorem 3.1.1 Let G be an SCCN-group.Then(1)If N(?)G,then G/N is an SCCN-group.And if N is a normal self-centralizing subgroup of G,then G/N is a CN-group.(2)G is a solvable group.Conversely,if G is a solvable group,and the C-normality of G is of transitivity in G,then G is an SCCN-group.(3)If Φ(G)≠1,then nl(G)≤2.In section 3.2,we discuss NSST-groups.And some conclusions have been obtained:Definition 3.2.1 Let G be a finite group.If every non-abelian self-centralizing subgroup of G is a subnormal subgroup or a TI-subgroup,then we call G is a NSST-group.Theorem 3.2.1 Let G be a NSST-group.Then every non-abelian subgroup of G is subnormal in G.Theorem 3.2.2 Let G be a NSST-group.Then G is a group of one of the following types:(1)G is nilpotent;(2)G=N(?)M is a Frobenius group with a kernel N and a complement M,and N is a minimal normal nilpotent subgroup of G and M is nilpotent.In section 3.3,we study the influence of the number of conjugate classes of self-centralizing subgroups on the solvability of finite groups.Let r(G)be the number of conjugate classes of self-centralizing subgroups of finite groups G.The results are as follows:Theorem 3.3.1 Let G be a finite group.Then r(G)=1 if and only if G is an abelian group.Theorem 3.3.3 Let G be a finite group.If r(G)≤5,then G ia a solvable group.Theorem 3.3.7 Let G be a finite group,and r(G)=3.Then one of the following holds:(1)G is a q-fundamental group,|G|=pαqβ,and q are prime numbers,and α and βare positive integers.There are two classes of conjugate of maximal subgroups in G,and one of which is normal and the other is non-normal.(2)G=Q(?)H,|Q|=qβ,(q,|H|)=1,H(?)G,H is nilpotent,and |Q,H|(?)Φ(G)for some maximal G-admissible subgroup Q of Q.(3)G=Q(?)H,|Q|=qβ(q,|H|)=1,H is nilpotent,and the image of[Q,H]in G/Φ(Q)is a minimal normal subgroup.(4)G=P(?)H,|P|=pα,(p,|H|)=1,H(?)G,H is nilpotent,and[P,H]Φ(P)/Φ(P)is a direct product of two non-H-isomorphic minimal H-admissible subgroups.(5)G=P(Q(?)H),|P|=pα,|Q|=qβ,(q,|H|)=1,H is nilpotent,P/P ∩Φ(G)is a noncentral minimal normal subgroup of G/P∩Φ(G),and[Q,H]Φ(Q)/Φ(Q)is a minimal normal subgroup of QH/Φ(Q).The fourth chapter includes a summary of the work done in this paper,as well as some prospects for this study. |
