Finite p-Groups all of Whose Subgroups of Index p^(3) are Abelian  

作　　者：Qinhai Zhang Libo Zhao Miaomiao Li Yiqun Shen 

机构地区：[1]Department of Mathematics,Shanxi Normal University,Linfen 041004,Shanxi,People’sRepublic of China

出　　处：《Communications in Mathematics and Statistics》2015年第1期69-162,共94页数学与统计通讯（英文）

基　　金：This work was supported by NSFC(Nos.11371232,11471198);by NSF of Shanxi Province(No.2013011001).

摘　　要：Suppose that G is a finite p-group.If all subgroups of index p^(t)of G are abelian and at least one subgroup of index p^(t−1)of G is not abelian,then G is called an A_(t)-group.We useA0-group to denote an abelian group.From the definition,we know every finite non-abelian p-group can be regarded as an A_(t)-group for some positive integer t.A_(1)-groups and A_(2)-groups have been classified.Classifying A_(3)-groups is an old problem.In this paper,some general properties about A_(t)-groups are given.A_(3)-groups are completely classified up to isomorphism.Moreover,we determine the Frattini subgroup,the derived subgroup and the center of every A_(3)-group,and give the number of A_(1)-subgroups and the triple(μ_(0),μ_(1),μ_(2))of every A_(3)-group,whereμi denotes the number of A_(i)-subgroups of index p of A_(3)-groups.

关 键 词：Finite p-groups Minimal non-abelian p-groups A_(t)-groups 

分 类 号：O15[理学—数学]