Some Studies On The Enumeration Problem Of Finite P-Groups

Groups of prime powers are called finite p-groups.Finite p-groups is an im-portant branch of finite groups.In recent years,there are more and more group theory scholars who pay their attention to finite p-groups.The enumeration prob-lem is a principal research subject of finite p-groups.And P.Hall made significant contributions to the research of enumeration problem.He proved that δk(G)≡0(mod pd-k+1),where G is a finite p-group,δk(G)denotes the number of subgroups of index pk which do not contain the Frattini subgroup of G,1 ≤k≤d=d(G),d(G)denotes the minimal cardinality of generating set of G.In this thesis,we re-search the enumeration problem of finite p-groups based on this consequence.There has been little subsequent research on this consequence in the past.The cases ofδk(G)=0 have been researched.So we study the case of δk(G)>0 in this thesis.And we always assume that 2 ≤k ≤d for the reason that all finite p-groups satisfyδ1(G)=0.This thesis consists of five chapters.We introduce the background of the enu-meration problem of finite p-groups in the first chapter.In the second chapter,we research finite p-groups with δk(G)=pd-k+1,and we get some properties and the classification of these groups.Based on these results we guess that the minimal val-ue of δk(G)depends on k when δk(G)>0.So in the third chapter,we calculate the minimal value of δk(G)when δk(G)>0.We discuss in two cases according to p>2 and p=2.And we get the quantitative relationship between the minimal value ofδk(G),k and d for p>2.For the case p=2,we obtain the quantitative relationship of the minimal value of δk(G)and k.These results confirm our conjecture.Next we want to know the possible value of δk(G).Since k≥2,we firstly consider the case k=2.And in the fourth chapter,we calculate the value of δ2(G)of finite p-groups with a cyclic subgroup of index p2.We discover that the range of the value of 62(G)is related to exp(G)by use of the classifications of finite p-groups with δk(G)=p and δd(G)=p2.In order to study the influence of exp(G)on the other arithmetical invariant of G,we classify finite p-groups with γ(G)≤13,and we investigate the quantitative relationship of exp(G)and γ(G)γ where γ(G)denote the number of conjugate class of noncyclic subgroups of G.