The Influence Of Weakly CAP*-Subgroups On The Structure Of Finite Groups

Let G be a finite group.A subgroup H of G is called a CAP*-subgroup of G,if H either covers or avoids every non-Frattini chief factor of G.A subgroup H of G is said to be a weakly CAP*-subgroup of G if there exists a normal subgroup K of G such that HK is normal in G,H? K is a CAP*-subgroup of G and ?(G/K)(?)?(H),? is a prime number set.In this paper,we investigate the structure of G under the assumption that certain subgroups of prime power orders are weakly CAP*-subgroups of G.In the investigation of finite groups,by using some properties of special subgroups to characterize the structure of finite group is a main method.The purpose of this thesis is to study the character of p-nilpotency and super solvability of finite group G by a weakly CAP*-subgroup,and we get some new characterizations of p-nilpotency and supersolvability of G:The full thesis is divided into two chapters according to contents:In the first chapter,we mainly introduce the investigative background,the preliminary notions and some relevant known results,the main properties and correlative lemmas which are related to weakly CAP*-subgroups.In the second chapter,we use some properties of weakly CAP*-subgroup to obtain some sufficient conditions for finite group G to be p-nilpotent or supersolvable.We obtain some main results as follows:(1)Let H be a normal subgroup of a group G such that G/H is p-nilpotent,p a prime dividing the order of H with gcd(|G|,p-1)= 1,and let P be a Sylow p-subgroup of H.If all maximal subgroups of P are CAP*-subgroups of G,then G is p-nilpotent.In particular,G is p-supersolvable.(2)Let G be a p-solvable group,p is a fixed prime dividing the order of G.Then the following statements are equivalent:(i)G is p-supersolvable.(ii)C has a normal subgroup H such that G/H is p-supersolvable and for a Sylow p-subgroup of H all maximal subgroups of P are weakly CAP*-subgroups of C.(3)Let p be a fixed prime dividing the order of a group G and let P be a Sylow p-subgroup of G.Then the following statements are equivalent:(i)G is p-supersolvable.(ii)P is a CAP*-subgroup of G and all maximal subgroups of P are weakly CAP*-subgroups of G.(4)Let G be a p-solvable group,p is a fixed prime dividing the order of G.Then the following statements are equivalent:(?)G is p-supersolvable.(?)G has a normal subgroup H such that G/H is p-supersolvable and for a Sylow p-subgroup of H,all maximal subgroups of Fp(H)are weakly Op'(H)?subgroups of G.(5)Let F be a saturated formation containing the class of all supersolvable groups U.Suppose that G is a group with a normal subgroup H such that G/H E,F.If all maximal subgroups of any Sylow subgroup of H are weakly CAP*-subgroups of G,then G E ?F.(6)Let F be a saturated formation containing the class of all supersolvable groups U.Suppose that G is a group with a normal subgroup H such that G/H E.F.If all maximal subgroups of any Sylow subgroup of F*(H)are weakly CAP*-subgrolps of G,then G ? F.