Some Prime-Power Order S-Quasinormally Embedded Subgroups And The Structure Of Finite Groups

A subgroup H of a group G is called S-quasinormally embedded in G if every Sylow subgroup of H is a Sylow subgroup of some S-quasinormally subgroup of G. In this paper, by using Sylow subgroups, we investigate the influence of S-quasinormally embedded subgroups on p-supersolvability and supersolvability of finite groups and the structure of the group.The paper is divided into two chapters. In the first chapter, we introduce the in-vestigative background, the preliminary notions, correlative lemmas and Theorems. In the second chapter, we use the properties of the S-quasinormally embedded subgroups to investigate the structure of finite groups, and obtain some sufficient conditions for a finite group to be p-supersolvable and supersolvable. We obtain some main results as follows:Theorem2.1.1Let p be a prime, and let H be a normal subgroup of such that G/H∈up. If the maximal subgroups of the Sylow p-subgroup P is S-quasinormally embedded in G, and if, in addition, H is p-soluble with p-length at most one, Then G∈up.Theorem2.2.1Let P be normal elementary abelian p-subgroup of a group G. If there exists a subgroup D of P such that1<|D|<|P|and every subgroup H of P with order|D|is S-quasinormally embedded in G, then every chief factor of G contained in P is cyclic.Theorem2.2.2Let F be a saturated formation containing U. and E be a normal subgroup of a group G 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|whenever|D|=2is S-quasinirmally embedded in G,then G∈F.