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On Some Embedded Subgroups and Supplemented Subgroups of Finite Groups

Abid Mahboob  
【摘要】:Throughout the thesis, all the groups considered are finite. The main purpose of this thesis is to investigate the structure of finite groups by using some embedded sub-groups, supplemented and CAP-subgroups. This thesis consists of five chapters as; Chapter I shows the motivation and the historical background of the problems which has been faced in the present thesis. In chapter Ⅱ, we introduce some basic concepts and notions in detail. In addition, a brief literature survey also included. Chapter Ⅲ is devoted to study the influence of S-semiembedded subgroups on the structure of finite groups. We give the new characterization of nilpotency and solubility of a group under the assumption that the maximal subgroups of Sylow subgroups are S-semiembedded (Theorem3.1,3.2). Further, we associate this idea with choice of the normalizer of Sylow p-subgroup (Theorem3.5,3.7). In the end, the applications of these results are discussed. In chapter Ⅳ, we introduce QΦ-supplemented subgroup and use it to investigate the structure of finite groups. We discuss the basic properties of QΦ-supplemented subgroups and use the QΦ-supplemented properties of some minimal subgroups to s-tudy the structure of a finite group. In particular, we give the necessary and sufficient of a group to be p-nilpotent and we discuss about the supersolubility of a group by the QΦ-supplemented of non cyclic Sylow p-subgroups. In chapter Ⅴ, we analysis the influence of semi cover-avoiding property or S-quasinormally embedded subgroups on the structure of finite groups. The idea is to characterize the nilpotency and supersolvability of a finite group G by assuming some subgroups of order with prime power either semi cover-avoiding or S-quasinormally embedded in G. Moreover, we assume maximal subgroups of Sylow subgroups either semi cover-avoiding or S-quasinormally embedded subgroups to obtain some new re-sults such that a group is supersoluble or belongs to some saturated formation under some assumptions (Theorem2.7,2.9). A series of new results has been obtained.


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