Study On Abelian Subgroups And The Structure Of Groups

In this paper we will study how abelian subgroups influent the structure of finite groups,there are four sections in our results.In the first section,namely 3.1,we obtained some description of the structure of some finite groups whose centralizers or normalizers of abelian subgroups satisfy some conditions.In the second section,namely 3.2,we consider some abelian subgroups whose centralizers are equal to its normalizers,so we obtain some sufficient conditions of p-nilpotent groups and p-closed groups.In the third section,namely 3.3,we consider some abelian subgroups,such as abelian subgroups generated by two elements,elementary abelian subgroups,maximal abelian subgroups,cyclical subgroups, minimal subgroups,whose centralizers are equal to its normalizers,so we obtain some necessary and sufficient condions of abelian groups and cyclic groups,and improve Zassenhaus Theorem and Chen Chongmu,Theorem 0.3 in Reference[2].In the last section,namely 3.4,we mainly discuss how abelian subgroups influent the solvability of finite groups, so we obtain some sufficient conditions of solvable groups.