On Weakly SS-Quasinormal Subgroups With Prime-Power Orders Of Finite Groups

Let H be a subgroup of a finite group G. We say that H is weakly SS-quasinormal if there exists a subgroup B of G such that HB is normal in G and for any prime p with gcd(p,|H|)=1, H permutes with every Sylow p-subgroup of B and Sylp(B)(?) Sylp(G). Here, H is also said to be a weakly SS-quasinormal subgroup of G.In the investigation of finite groups, using some properties of subgroups to char-acterize the structure of finite group G is a effective method. The purpose of this paper is to study the influence of weakly SS-quasinormal subgroups on the struc-ture of finite groups such as p-nilpotency and supersolvability. The paper is divided into two chapters. In the first chapter, we introduce the investigative background, the preliminary notions, correlative lemmas and theorems. In the second chapter, we use the properties of the weakly SS-quasinormal subgroups to investigate the structure of finite groups, and obtain some sufficient conditions for a finite group to be p-nilpotent and supersolvable. We obtain some main results as follows:Lemma2.1.1Let H be a nilpotent subgroup of G. Then the following statements are equivalent:(1) H is S-quasinormal in G.(2) H≤F(G) and H is weakly SS-quasinormal in G.Lemma2.1.2Let p be the smallest prime dividing the order of G, and P∈Sylp(G). If every maximal subgroup is weakly SS-quasinormal in G, Then G is p-nilpotent.Lemma2.1.3Let P be a normal elementary abelian p-subgroup of a group G. If there exists a subgroup D in P such that1<|D|<|P|and every subgroup H of P with|H|=|D|is weakly SS-quasinormal in G,then every subgroup of P with order p is S-quasinormal in G.Lemma2.1.4Let P be a normal elementary abelian p-subgroup of a group G. If there exists a subgroup D in P such that1<|D|<|P|and every subgroup H of P with；|H|=|D|is weakly SS-quasinormal in G,then every chief factor of G contained in P is cyclic.Theorem2.2.1Let F be a saturated formation containing U and G be a group with a normal subgroup E such that G/E∈F.If,for every non-cyclic Sylow subgroup P of E,there is a subgroup D of P with1<|D|<|P|such that every subgroup of P with order|D|or2|D|is weakly SS-quasinormal in G,then G∈F.Theorem2.2.2Let p be a prime dividing the order of G and P a Sylow p-subgroup of G.If NG(P)is p-nilpotent and P has a subgroup D such that1<|D|<|P|and every subgroup H of P with order|H|=|D|,if p=2,every cycolc subgroup of P of order2or4is weakly SS-quasinormal in G or G is Q8-free,then G is p-nilpotent.Theorem2.2.3Let p be a prime dividing the order of G with gcd(p-1,|G|)=1and P is a Sylow p-subgroup D such that1<|D|<|P|and every subgroup H of P with order|H|=|D|is weakly SS-quasinormal,if P(?)G,then G is p-nipotent；if P(?)G,then G is p-closed group.if p=2,every cyclic subgroup of P of order2or4is weakly SS-quasinormal in G or G is Q8-free,then G is p-nilpotent.Theorem2.2.4Let F be a saturated formation containing U and G be a group with a normal subgroup E such that G/E∈F.Then G∈F if and only if for every noncyclic Sylow subgroup P of F’(E),there is a subgroup D such that1<|D|<|P|and every subgroup H of P with order|H|=|D|is weakly SS-quasinormal in G,and if,in addition,every cyclic Subgroup of P of order2or4is weakly SS-quasinormal in G or G is Q8-free.