The Influence Of Nearly CAP~*-Subgroups On The Structure Of Finite Groups

Let G be a finite group.A subgroup H of G is said to be 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 nearly CAP*-subgroup of G,if G has a subnormal subgroup K such that HK=G and H? K is a CAP*-subgroup of G.Let p be a prime divisor of |G|,P a Sylow p-subgroup of G.We denote by M(P),the set of all maximal subgroups of P.Let d be the minimum number of generators of P,denoted by Md(P)={P1,P2,�,Pd} such that?i=1dPi=?(P).In this thesis,we assume that some certain subgroups of prime power orders are nearly CAP*-subgroups to characterize the structure of G.In the investigation of finite groups,it's widerly applied to characterize the structure of finite groups by studying some special subgroups.In this thesis,we introduce a new subgroup called a nearly CAP*-subgroup of G to study the structure of finite groups.Some related results are generalized and some new characterizations are obtained for a finite group to be p-nilpotent or supersolvable.The main contents of this thesis are divided into two chapters.In the first chapter,we present the basic definitions,the background and relevant known results,and introduce the lemmas needed in the study.In the second chapter,we mainly study how the nearly CAP*-subgroups influence the structure of finite groups,and some necessary and sufficient conditions for a finite group to be p-nilpotent or supersolvable are obtained.The main results are as follows:Theorem 2.1.1 Let G be a finite group,H a normal subgroup of G such that G/H is p-nilpotent and P a Sylow p-subgroup of H,where p is a prime factor of |G| with(|G|,p-1)=1.If all maximal subgroups of P are nearly CAP*-subgroups of G,then G is p-nilpotent.In particular,G is p-supersolvable.Theorem 2.1.2 Let G be a finite group,P a Sylow p-subgroup of G,and p be the minimal prime dividing |G|.Then the following statements are equivalent:(1)G is p-nilpotent(In particular,G is p-supersolvable).(2)Every maximal subgroup of P is a nearly CAP*-subgroup of G.Theorem 2.1.3 Let G be a finite group,and p be a prime factor of |G| with(|G|,p-1)=1.Assume that P is a Sylow p-subgroup of G,Then the following statements are equivalent:(1)G is p-nilpotent.(2)Each member of M(P)is a nearly CAP*-subgroup of G.(3)Each member of Md(P)is a nearly CAP*-subgroup of G.Theorem 2.2.1 Let G be a finite group,and let p be a prime factor of |G|.Then the following statements are equivalent:(1)G is p-supersolvable.(2)There exits a p-solvable normal subgroup H of G such that G/H is p-supersolvable,and every maximal subgroup of Sylow p-subgroup P of H is a nearly CAP*-subgroups of G.Theorem 2.2.2 Let G be a finite group,p a prime factor of |G|,and P a Sylow p-subgroup of G.Then the following statements are equivalent:(1)G is p-supersolvable.(2)P and all maximal subgroups of P are nearly CAP*-subgroups of G.Theorem 2.2.3 Let G be a p-solvable group,H a normal subgroup of G such that G/H is p-supervable,where p is a prime.If all maximal subgroups of Fp(H)containing Op'(H)are nearly CAP*-subgroups of G,then G is p-supersolvable.Theorem 2.2.4 Let F be a saturated formation containing u,G a finite group and H a normal subgroup of G such that G/H ? F.If all maximal subgroups of Sylow-subgroups of H are nearly CAP*-subgroups of G,then G ?F.