The Normalizers Of Subgroups And The Structure Of Finite P-groups

In the study of the finite group theory, the normality of subgroups and the permutability between subgroups are basic starting points to study finite groups. We know that normalizers and centralizers of subgroups are the measurement of the normality and the abelity of subgroups. So to study the structure of finite groups by normalizers and centralizers of subgroups becomes one of topics in the finite group theory, and lots of important results have been obtained. In this paper, we use normalizers and centralizers of subgroups to study the structure of finite p-groups. Moreover, we also investigate the structure of a finite 2-group G with r(G)=2.In chapter III, we investigate the structure of finite 2-equilibrated p-groups. A finite group G is said to be n-equilibrated provided that for all subgroups H and K of G with d(H)> n and d(K)> n, either H< NG(K) or K< NG(H). We provide a complete classification of such groups when G is 2-generator. If G is 3-generator, we prove that G has a normal metacyclic subgroup N such that G/N is cyclic and, if G has at least four generators, we prove that G is modular and G’< (x) for any element x in G with o(x)= exp(G).In chapter IV, we are interested in the structure of CAC-p-groups. A finite p-group G is called a C4C-p-group if Ca(H)/H is cyclic for every non-cyclic abelian subgroup H in G with H(?)Z(G). We give a complete classification of finite CAC-p-groups.In chapter V, we study the finite 2-group G in which r(G)=2 and G has more than three involutions. We prove that if Ω1(G)≌D2n or D2n*C4 with n≥3, then G’ is abelian and there exists a maximal subgroup M of G such that M is metacyclic.If Ω1(G)≌D2n*Q2m with n,m≥3,then either Φ(G)≤Ω1(G), or |Ф(G)|=|Ω1(G)| and G’∩Ω1(G) is a maximal subgroup of Ω1(G).