Supplemented Or Permutable Subgroups And The Structure Of Finite Groups

The aim of this Ph.D. dissertation is to study the structure and properties of finitegroups in which subgroups are supplemented or permutable.In Chap I, we introduce some research background and some main results of thisdissertation.In Chap II, we introduce the notations and terminologies used in this paper.In Chap III, we study the influence of supplemented subgroups on the structure offinite groups. First, we give the structure of finite groups in which Sylow normalizershave nilpotent Hall supplements (Theorem 3.2.3, 3.2.5 and 3.2.8). Secondly, we es-tablish some new G-covering subgroup systems by supplemented subgroups (Theorem3.3.1). Thirdly, we use F-supplemented subgroups to obtain some new criteria of someclasses of groups (Theorem 3.4.4, Theorem 3.4.13-3.4.15 and Theorem 3.4.19).In Chap IV, we research into the structure of finite groups with S-C-permutablyembedded subgroups and give some new characterizations of some classes of finitegroups (Theorem 5.2.1-4.2.3, Theorem 4.2.7, Theorem 4.2.9-4.2.10).In Chap V, based on the analysis of some previous researches, we give the conceptof F-quasinormal subgroups and study its in?uence on the structure of finite groups.first, we get some new descriptions of supersoluble groups by using the minimal sub-groups, the maximal subgroups of Sylow subgroups, Fitting subgroups and generalFitting groups (Theorem 5.2.1-5.2.2, Theorem 5.2.4, Theorem 5.2.6-5.2.7). Secondly,we use the maximal subgroups and Sylow subgroups to give two necessary and suf-ficient conditions of soluble groups (Theorem 5.2.8-5.2.9). Thirdly, we obtain somenew criteria of p-nilpotent groups by i-minimal subgroups of a group and i-maximalsubgroups of Sylow subgroups (Theorem 5.2.10-5.2.13) of a group. A series of knownresults were generalized.All groups in this paper are finite. The research results in this paper are used thetheory and the method of the class of groups.