The Influence Of Semi-CAP-subgroups And C~#-normal Subgroup On The Structure Of Finite Groups

Group theory is an important branch of algebra.Many scholars have been working on it and related topics.Thereinto,the structure of groups is one of import contents.One of the effective methods in the study of structure of groups is the use of properties of subgroups.Since 1939,some scholars have generalized the definition of normal subgroups.There are more and more scholars working on the structure of groups by use of properties of subgroups,and the related papers are more and more abundant.This paper deals with the structure of groups by using the semi-cover-avoidance and c#-normal properties of subgroups.On the one hand,we use the semi-cover-avoiding of maximal subgroups of Sylow subgroup(semi CAP-subgroup)to describe the structure of finite groups,We obtain some new sufficient and necessary conditions for a group to be p-supersolvable,p-nilpotent and supersolvable.And two main theorems can be achieved:(1)Let G be a group and let P be a Sylow p-subgroup of G,where p ? ?(G).Suppose that every maximal subgroup of P is semi CAP-subgroup in G.Then,either G is p-supersolvable or P is of order p.(2)Let A be a s-quasinormal subgroup of G,B be a subgroup of G and G = AB.Then G is p-nilpotent if and only if every maximal subgroup of A and B Sylow p-subgroup is semi CAP-subgroup of G,where p ? ?(G)and(|G|,p-1)= 1.On the other hand,we use the c#-normality of maximal subgroups of Sylow sub-group,and present some new sufficient and necessary conditions for a group to be p-supersolvable,p-nilpotent and supersolvable by reducing the number for maximal subgroups of Sylow subgroup.The following main theorems are obtained.(1)Let G be a p-solvable group and let P be a Sylow p-subgroup of G,where p is a prime divisor of |G|.Then G is p-supersolvable if and only if every member in some fixed Md(P)is c#-normal in G.(2)Let G be a group and let P be a Sylow p-subgroup of G,where p is a prime divisor of |G|.Suppose that every maximal subgroup of P is c#-normal in G.Then,either G is p-supersolvable or P is of order p.Through the study of this paper,our results generalize some results on c-normality and semi-cover-avoiding.