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finite p-groups with Few Non-major k-Maximal Subgroups

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Chinese Annals of Mathematics,Series B 2018年第1期39卷 59-68页

作者： Boyan WEI Haipeng QU Yanfeng LUO School of Mathematics and Statistics Lanzhou University School of Mathematics and Computer Science Shanxi Normal University

A subgroup of index p^k of a finite p-group G is called a k-maximal subgroup of *** by d(G) the number of elements in a minimal generator-system of G and by δ_k(G) the number of k-maximal subgroups which do not contain the Frattini subgroup of *** this paper,the authors classify the finite p-groups with δ_(d(G))(G) ≤ p^2 and δ_(d(G)-1)(G) = 0,respectively.

关键词： finite p-groups k-Maximal subgroups k-Major subgroups Frattini subgroup The number of non-major k-maximal subgroups

finite p-groups whose non-normal subgroups have few orders

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Frontiers of Mathematics in China 2018年第4期13卷 763-777页

作者： Lijian AN Department of Mathematics Shanxi Normal University Linfen 041004 China

Suppose that G is a finite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use p^M（G） and p^m（G） to denote the maximum and minimum of the orders of the non-normal subgroups of G, respectively. In this paper, we classify groups G such that M（G）〈 2m（G） ^- 1. As a by-product, we also classify p-groups whose orders of non-normal subgroups are p^k and p^k＋1.

关键词： finite p-groups meta-hamiltonian p-groups non-normal subgroups

finite p-groups all of Whose Maximal Subgroups Either are Metacyclic or Have a Derived Subgroup of Order ≤ p

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Chinese Annals of Mathematics,Series B 2015年第1期36卷 11-30页

作者： Lihua ZHANG Yanming XIA Qinhai ZHANG Department of Mathematics Beijing University of Posts and Telecommunications Department of Mathematics Shanxi Normal University

The groups as mentioned in the title are classified up to isomorphism. This is an answer to a question proposed by Berkovich and Janko.

关键词： finite p-groups Nonmetacyclic p-groups Minimal nonabelian p-groups Maximal subgroups

finite p-groups all of Whose Subgroups of Index p^(3) are Abelian

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Communications in Mathematics and Statistics 2015年第1期3卷 69-162页

作者： Qinhai Zhang Libo Zhao Miaomiao Li Yiqun Shen Department of Mathematics Shanxi Normal UniversityLinfen 041004ShanxiPeople’sRepublic of China

Suppose that G is a finite *** 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)-*** useA0-group to denote an abelian *** 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 *** A_(3)-groups is an old *** this paper,some general properties about A_(t)-groups are given.A_(3)-groups are completely classified up to ***,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

Some unsolvable conjectures in finite p-groups

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Frontiers of Mathematics in China 2022年第1期17卷 1-22页

作者： Qinhai ZHANG Department of Mathematics Shanxi Normal UniversityTaiyuan 030031China

We survey some unsolvable conjectures in finite p-groups and their research progress.

关键词： finite p-groups Hua-Tuan’s conjecture Higman’s conjecture Oliver’s conjecture Wiegold’s conjecture

Semicomplete finite p-groups of Class 2

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Algebra Colloquium 2016年第4期23卷 651-656页

作者： M. Shabani Attar Department of Mathematics Payame Noor University P.O. Box 19395-4697 Tehran Iran

Let G be a group and G＇ be its commutator subgroup. An automorphism α of a group G is called an IA-automorphism if x^-1α（x） ∈G＇ for each x∈G. The set of all IA-automorphisms of G is denoted by IA（G）. A group G is called semicomplete if and only if IA（G） = Inn（G）, where Inn（G） is the inner automorphism group of G. In this paper we completely characterize semicomplete finite p-groups of class 2; we also classify all semicomplete finite p-groups of order p^n （n≤5）, where p is an odd prime. This completes our work in 2011.

关键词： semicomplete groups IA-automorphisms finite p-groups

finite p-groups with a Cyclic Subgroup of Index p3

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Journal of Mathematical Research with Applications 2012年第5期32卷 505-529页

作者： Qinhai ZHANG,pujin LI Department of Mathematics,Shanxi Normal University,Shanxi 041004,p.R.China Department of Mathematics Shanxi Normal University

We classify up to isomorphism those finite p-groups,for odd primes p,which contain a cyclic subgroup of index p3.

关键词： finite p-groups inner abelian p-groups metacyclic p-groups regular p-groups irregular p-groups.

A Class of finite Resistant p-groups

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Acta Mathematica Sinica,English Series 2017年第5期33卷 725-730页

作者： He Guo LIU Yu Lei WANG Department of Mathematics Hubei University Department of Mathematics He'nan University of Technology

A finite p-group p is called resistant if, for any finite group G having p as a Sylow p-group,the normalizer N_G(p) controls p-fusion in G. Let p be a central extension as 1→ Z_(p^m)→ p→ Z_p × · · · × Z_p→1,a... 详细信息

关键词： finite p-groups symplectic groups fused p-centric

finite 2-groups whose length of chain of aonnormal subgroups is at most 2

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Frontiers of Mathematics in China 2018年第5期13卷 1075-1097页

作者： Qiangwei SONG Qinhai ZHANG Department of Mathematics Shanxi Normal University Linfen 041004 China

Assume that G is a finite non-Dedekind p-group. D. S. passman introduced the following concept： we say that H1 〈 H2〈.. 〈 Hk is a chain of nonnormal subgroups of G if each Hi G and if ｜Hi ： Hi-1｜ = p for i = 2, 3,..., k. k is called the length of the chain, chn（G） denotes the maximum of the lengths of the chains of nonnormal subgroups of G. In this paper, finite 2-groups G with chn（G） ≤ 2 are completely classified up to isomorphism.

关键词： finite p-groups chain of nonnormal subgroups