The Influence Of SS-Supplemented Subgroups On The Structure Of Finite Groups

Let G be a finite group, a subgroup H of a group G is said to be S-quasinormal subgroup in G, if H permutes with every Sylow subgroup of G; a subgroup H of a group G is said to be SS-quasinormal subgroup in G, if there exists a subgroup K of G such that G= HK and H permutes with every Sylow subgroup of K; a subgroup H of a group G is said to be SS-supplemented subgroup in G if there exists a subgroup K of G such that G=HK and H ∩ K is S-quasinormal in K.In the investigation of finite groups, by using some properties of special subgroups to characterize the structure of finite group is a main method. In this thesis, by using the properties of S-quasinormal subgroup and SS-supplemented subgroup, we investigate the characterization of p-nilpotency and supersolvability of G, and we get some new charac-terizations of p-nilpotency and supersolvability of G. The full thesis is divided into two chapters according to contents:In the first chapter, we mainly give the new concept of S-quasinormal subgroup and SS-supplemented subgroup, introduce the investigative back-ground, the preliminary notions and some relevant known results, the main properties and correlative lemmas which are related to S-quasinormal subgroup and SS-supplemented sub-group. In the second chapter, by using some properties of the S-quasinormal subgroup and SS-supplemented subgroup, we get some sufficient conditions for a finite group G to be p-nilpotent and supersolvable. We obtain some main results as follows:Theorem 2.1.1 Let G be a finite group and p the smallest prime divisor of|G| and let P be a sylow p-subgroup of G. If P is A4-free and all minimal subgroups of D(G) ∩ P are SS-supplemented in G, then G is p-nilpotent.Theorem 2.1.2 If all minimal subgroups of D(G) D P are SS-supplemented in G for all Sylow subgroups P of G, then G is supersolvable or G has a section isomorphic to the quaternion group of order 8.Theorem 2.1.3 Let F be the class of groups with Sylow tower of supersolvable type and let H be a normal subgroup of G such that G/H F , p is a prime divisor of|H|, where P is a Sylow p-subgroup of H. If every minimal subgroup of D(H) ∩ P is SS-supplemented in H, then G € F.Lemma 2.2.1 Let G be a group and p be the smallest odd prime divisor of|G|. If G has a proper subgroup H with index p, then G/HG is solvble.Theorem 2.2.1 Let G be a group and p be any odd prime divisor of|G|. If every minimal subgroup of G’∩ P is SS-supplemented in G, where P is a sylow p-subgroup of G, then G is solvable.Theorem 2.2.2 Let G be a group. If every subgroup of G with order 3 and 5 is SS-supplemented in G, then G is solvable.Theorem 2.2.3 Let G be a group. If G be a non-solvable groups, Then lo(G)≥|π(G)|. In particular, lo(G)=|πn(G)|if and only if G≌ A5 or SL(2,5).