On Some ?-properties Of Subgroups Of Finite Groups And The Structure Of (?)_?-critical Groups

In this Ph.D.dissertation,we study the structure of G under the condition that some given subgroups of G are weakly ?-permutable(resp.?-permutably embedded,n-?-embedded)in G and the structure of(?)?-critical groups.Throughout this dissertation,all groups are finite.This dissertation is divided into six chapters.In Chapter ?,we introduce the background of the research and the main results of this dissertation.In Chapter ?,we list some basic terminologies and some known results which will be used in this dissertation.In Chapter ?,we study the relationship between weakly ?-permutable subgroups and the structure of finite groups.We first introduce the concept of weakly ?-permutable subgroups.Some new characterizations about ?-solubility and supersolubility are ob-tained by studying the structure of finite groups under the condition that some given subgroups are weakly ?-permutable(see Theorem 3.2.1 and Theorem3.2.3).In partic-ular,we give the conditions under which a normal subgroup of G is hypercyclically embedded(see Theorem 3.2.5).Some known results are generalized(see Corollaries 3.3.1-3.3.14).In Chapter IV,we study the influence of ?-permutably embedded subgroups on the structure of finite groups.We obtain a new characterization about supersolubility by studying the ?-permutably embedded property of maximal subgroups of subgroups which in a complete Hall ?-set(see Theorem 4.2.2).In addition,some new criteria for a normal subgroup is hypercyclically embedded are obtained(see Theorem 4.2.3 and Theorem 4.2.5).In Chapter V,we introduce the new concept of n-?-embedded subgroups,and dis-cuss their influence on the structure of finite groups.We establish some new criteria for a group to be ?-soluble or supersoluble or more general,a group belongs to a saturated formation containing all supersoluble groups(see Theorem 5.2.1,Theorem 5.2.3 and Corollaries 5.3.1-5.3.4).These results unify and generalize some known achievements.In Chapter VI,we study the structure of(?)?-critical groups.We first prove that every(?)?-critical group is ?-soluble.This result gives a positive answer to a recent open problem of A.N.Skiba.We also prove that(?)?-critical groups are also Schmidt groups and so the structure of(?)?-critical groups is obtained.