Weakly C-Normal Subgroup Of Finite Group And It's Generalization

A subgroup H of a finite group G is said to be weakly c-normal in G if there exists a subnormal subgroup K of G such that G = HK and H∩K ≤ HG, where Hq is the maximal normal subgroup of G that is contained in H. The study about weakly c-normal subgroups has interested scholars for a long time, and the research in this field is still very active. In this paper, the structure of groups are investigated by means of c-normal subgroups, and some new characterizations about the structure for a group are proved. We also imply and generalize some known theorem by use of the concept of weakly c-normal subgroup.The thesis is divided into four sections according to contents.In chapter 1, the relationship between the c-normal subgroup and the weakly c-normal subgroup was studied. A subgroup H of a finite group G is said to be c-normal in G if there exists a normal subgroup K of G such that G = HK and H ∩K≤ Hg, where HG is the maximal normal subgroup of G that is contained in H. It is easy to see from the definition that a c-normal subgroup of G must be weakly c-normal subgroup, but the converse is not ture ingeneral. In this chapter, we gave some sufficient conditions for a weakly c- normal subgroup to be c-normal. Using these conditions, some known theorems and properties about c-normal subgroup were generalized by use of weakly.c-normal subgroup.In chapter 2, by using the weakly c-normality of Sylow subgroups and π-Hall subgroup, some sufficient conditions of solvable groups were obtained. In addition, we consider the relationship between the normal index of maximal subgroup and weakly c-normal subgroup. Deskin introduced the concept of normal index of a maximal subgroup. The normal index of a maximal subgroupM of G, denoted by rj(G : M), is defined as the order of a chief factor H/K of G, where H is a minimal supplement to M in G. The investigations on the normal index have been developed by some scholars. Deskin showed that G is solvable if and only if r)(G : M) = \G : M\ for every maximal subgroup M of G. Yanming Wang proved a maximal subgroup M of G is c-normal in G if and only if r](G : M) = \G : M\. Similarly, there exist some relationships between weakly c-normality and the normal index. It is showed in this chapter that a maximal subgroup M of G satisfy -q{G : M) = \G : M\ if and only if M is solvable and M is weakly c-normal in G. In addition, we generalized an important theorem for solvable group:a group G is solvable if and only if G has a solvable maximal subgroup M such that M is weakly c-normal in G.In chapter 3, the supersolvability of groups were proved. The minimal subgroups plays an important role in the study of finite groups. In this chapter, we use the weakly c-normality of minimal subgroups to characterize the structure of the groups, obtain some sufficient conditions under which a group belongs to supersolvable group. We proved that if the minimal subgroup and the cyclic subgroup of order 22 of G is weakly c-normal in G, then G is super-solvable.In the last chapter, s-normal subgroup, a concept which is weaker than weakly c-normal subgroup was introduced and investigated. A subgroup H of a finite group G is said to be s-normal in G if there exist a subnormal subgroup K of G such that G = HK and H n K < Hsg, where Hsg is the maximal subnormal subgroup of G that is contained in H. Some properties of s-normal subgroups were given.In addition, by using the s-normality of subgroups, we obtain new sufficient conditions for solvability of groups.