The Embedding Of Prime-power-order Subgroups And The Structure Of Finite Groups

In finite group theory, studying the structure of finite groups through the embedding property of subgroups is an interesting subject, Since the normality and the complementation of subgroups are the most elementary properties in finite group theory, there are many embedding properties of subgroups based on them, and a lot of results have been obtained. Following that, our paper is focused on the embedding property of subgroups of prime power order, and many new results are also obtained.In chapter ?, we obtain some sufficient conditions for some normal subgroup of a finite group G to be contained in the F-hypercentre of G by using some family of weakly M-supplemented subgroups of prime power order, where T is a solvable saturated formation containing U.In chapter ?, we get some sufficient conditions for some normal p-subgroup of a finite group G to have every G-chief factor below it cyclic by studying the S-semipermutability of noncyclic subgroups of order d of OP(G),where d is a power of some prime p.In chapter ?, we first introduce a new embedding property of subgroups,which is Il*-property, then prove that some normal subgroup E of a finite group G is p-hypercyclically embedded in G if and only if some classes of p-subgroups of E have II*-property in G.In chapter ?, we investigate the Ap-1-residual of a finite group G, that is GAp-1.where Ap-1 is the class of abelian groups with exponent dividing p-1.We not only give the elementary property of GAp-1, but also obtain some criteria for p-supersolvability of G by studying the particular embedding property of subgroups of given order of Op(GAp-1).