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

The essential task of finite groups is to determine the structure of finite groups.It is very interesting to study the structure of finite groups by using prime-power-order subgroups satisfying some embedding properties,and lots of important results have been obtained,for example Frobenius's Theorem and Glauberman-Thompson's Criterion for p-nilpotent.In this paper,we study the structure of finite groups by reducing the number of prime-power-order subgroups,and obtain some new results.In chapter III,We study the structure of finite groups mainly by using non-cyclic prime-power-order subgroups satisfying some embedding properties.First,we use non-cyclic p-subgroups H of order p^e satisfying H ? O^p?G????O^p?G?or H?O^p(G_???p-1)???O^p?G?to investigate the structure of finite groups.More-over,we study the structure of finite groups if non-cyclic p-subgroups H are S-semipermutable or satisfy condition H ? O^p?G????O^p?G?.In chapter IV,we mainly study the relationship between the structure of finite groups and the embedding property of non-maximal class prime-power-order subgroups.We first study the related property of non-maximal class sub-groups.Then we investigate the structure of finite groups if its non-maximal class prime-power-order subgroups are 3_p-permutable and satisfy the embedding H? O^p?G????O^p?G?.In chapter V,we study the structure of finite groups if every non-metacyclic subgroups of order pe satisfy the embedding H ? O^p(G^???_p2-1)???O^p?G?.