A Classification Of Finite P-groups With Small C_k(G)

Let G be a group of order pn.Notation ck(G)denotes the number of cyclic subgroups of order pk of G,P1 the finite p-groups satisfying "ck(G)≤ 2p for all k≥2",P2 the finite p-groups satisfying "ck(G)≤p2 for all k≥2" and P3 the finite p-groups satisfying"c2(G)≤ p2".In this thesis,we completely classify the P2-groups and P3-groups for p>2.This thesis is divided into five chapters.The first chapter is the introduction,which mainly introduces the research back-grounds,research methods and main results of this thesis.The second chapter is the preliminaries,mainly introduces the definitions and lemmas to be used in this thesis,and proves some lemmas to be used in this thesis.The third chapter gives the classification of finite P2-groups,which is divided into cases regular and irregular.For the regular P2-groups,the classification results are not difficult to obtain by the unique bases of the regular p-groups.For the irregular P2-groups,where p>2,we firstly prove that p=3 in this case,and then pick out the irregular P2-groups from the groups table of 34 order groups and 35 order groups.Secondly,we give the classification of the inner regular P2-groups of order ≥36.Finally,we prove that the irregular P2-groups of order≥36 are inner regular groups through the method of cyclic expansion.In summary,the P2-groups are completely classified.As a corollary,the P1-groups are also classified.The theories and methods of regular p-groups play a important role in this classification,they simplify our calculations significantly.The fourth chapter give the classification of P3-groups for p>2 using the methods as to P2-groups.The result shows that P2=P3 when p>2.The fifth chapter introduces and explores some problems that can be further studied.