Weakly C-normal Subgroup Of A Finite Group And Its Nature

There has been much interest in the past in investigating the relation between the properties of maximal subgroups of finite group G and the structure of G. In this aspect, the concept of a c-normal subgroup in a finite group was introduced by Wang in [1] and he proved that a finite group is solvable if and only if M is weak c-normal in G for every maximal subgroup M of G. As an application of the above result, some known theorems were generalized by using the concept "c-normality". Thus, c-normality provides a useful tool for the investigation of the structure of finite groups. A subgroup H is called weak c-normal in a group G if there exists a normal subgroup N of G such that G=HN and H∩N≤HG In fact, the concept of c-normal subgroup is closely related to the concept of normal subgroup. In this paper, we replace the N normality with N subnormality in c-normal concept to get the weak c-concept. We generalize Theorem 3.4,3.5 .in [1].In this paper, the author gives sufficient and necessary condition of solvable group using the weak c-normality of maximal and Sylow subgroup and gets some results about supersolvability and p-nilpotentlity of groups...