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Polycyclic Groups Admitting a Regular Automorphism of Order Four

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Acta Mathematica Sinica,English Series 2017年第4期33卷 565-570页

作者： Tao XU he guo liu Department of Science Hebei University of Engineering Handan 056038 P. R. China Department of Mathematics Hubei University Wuhan 430062 P. R. China

Let G be a polycyclic group and α a regular automorphism of order four of G. If the map φ: G→ G defined by g;= [g, α] is surjective, then the second derived group of G is contained in the centre of G. Abandoning t... 详细信息

关键词： Polycyclic group regular automorphism residually finite

The Isomorphism Problem of Normalized Unit Groups of Group Algebras of a Class of Finite 2-groups

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Acta Mathematica Sinica,English Series 2023年第11期39卷 2275-2282页

作者： Yu Lei WANG he guo liu Department of Mathematics He’nan University of TechnologyZhengzhou 450001P.R.China Department of Mathematics Hainan UniversityHaikou 570228P.R.China

Let p be a prime and F_p be a finite field of p *** F_(pG)denote the group algebra of the finite p-group G over the field F_(p)and V(F_(pG))denote the group of normalized units in F_(pG).Suppose that G and H are finite p-groups given by a central extension of the form 1→Z_(p)^(m)→G→Z_(p)×···×Z_(p)→1 and G'≌Z_(p),m≥*** V(F_(p)G)≌V(F_(p)H)if and only if G≌*** and Bovdi only solved the isomorphism problem when p is *** this paper,the case p=2 is determined.

关键词： isomorphism problem normalized unit Frattini subgroup finite 2-group

Automorphisms of a Class of Finite p-groups with a Cyclic Derived Subgroup

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Acta Mathematica Sinica,English Series 2021年第6期37卷 926-940页

作者： Yu Lei WANG he guo liu Department of Mathematics He’nan University of TechnologyZhengzhou 450001P.R.China Department of Mathematics Hubei UniversityWuhan 430062P.R.China

Let p be an odd prime,and let k be a nonzero nature *** that nonabelian group G is a central extension as follows1→G’→G→Z_(pK)×…×Z_(pK),where G’≌Zpk,andζG/G’is a,direct factor of G/G’.Then G is a central product of an extraspecial pkgroup E andζ***|E|=p(2n+1)k and|ζG|=p(m+1)*** that the exponents of E andζG are pk+l and pk+r,respectively,where 0≤l,r≤*** AutG’G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G’,let AutG/ζG,ζG G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the centerζG and let AutG/ζG,ζG/G’G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially onζG/G’.Then(ⅰ)The group extension 1→Aut G’→Aut G→Aut G’→1 is split.(ⅱ)AutG’G/AutG/ζG,ζG G≌G1×G2,where Sp(2n-2,Zpk)■H≤G1≤Sp(2n,Zpk),H is an extraspecial pk-group of order p(2n-1)k and(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)≤G2≤GL(m,Zpk)■Zpk(m).In particular,G1=Sp(2n-2,Zpk)■H if and only if l=k and r=0;G1=Sp(2n,Zpx)if and only if l≤r;G2=(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)if and only if r=k;G2=GL(m,Zpk)■Zpk((m))if and only if r=0.(ⅲ)AutG’G/Aut G/ζG,ζG/G’G≌G1×G3,where G1 is defined in(ⅱ);GL(ml,Zpk)■Zpk(m-1)≤G3≤GL(n,Zpk).In particular,G3=GL(m-1,Zpk)■Zpk(m-1)if and only if r=k;G3=GL(m,Zpk)if and only if r=0.(ⅳ)AntG/ζG,ζG/G’G≌AutG/ζG,ζG/G’G■Zpk(m),If m=0,then AntG/ζG,ζG/G’G=Inn G≌Zpk(2n);If m>0,then AntG/ζG,ζG/G’G≌Zpk(2nm)×Zpk-r(2n),and AutG/ζG,ζG G/Inn G≌Zpk((2n(m-1))×Zpk-r(2n).

关键词： Automorphism p-group cyclic derived subgroup symplectic group

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

Fusion Systems over M_p(1, 1, 1) × A-groups

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Acta Mathematica Sinica,English Series 2019年第8期35卷 1311-1320页

作者： Jun LIAO he guo liu Ji Ping ZHANG School of Mathematics and Statistics Hubei University Wuhan 430062 P. R. China School of Mathematical Sciences Peking University Beijing 100871 P. R. China

In this paper we study the saturated fusion systems over a direct product of the extraspecial group of order p^3 of exponent p and a finite abelian p-group. The result provides some new exotic fusion systems, whose un... 详细信息

关键词： Finite groups fusion systems generalized Fitting subgroups exotic fusion systems

The Automorphism Group of a Class of Nilpotent Groups with Infinite Cyclic Derived Subgroups

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Acta Mathematica Sinica,English Series 2018年第7期34卷 1151-1158页

作者： he guo liu Yu Lei WANG Ji Ping ZHANG Department of Mathematics Hubei University Wuhan 430062 P. R. China Department of Mathematics Henan University of Technology Zhengzhou 450001 P. R. China The School of Mathematical Sciences Peking University Beijing 100871 P. R. China

The automorphism group of a class of nilpotent groups with infinite cyclic derived subgroups is determined. Let G be the direct product of a generalized extraspecial E-group E and a free abelian group A with rank m, where E={{1 kα1 kα2…kαn aα＋1 0 1 0 … 0 αn＋2 0 0 0 … 1 α2n＋1 0 0 0 …0 1}}αi∈Z,i=1,2,…,2n＋1},where k is a positive integer. Let AutG＇G be the normal subgroup of AutG consisting of all elements of AutG which act trivially on the derived subgroup G＇ of G, and Autc G/ζG,ζGG be the normal subgroup of AutG consisting of all central automorphisms of G which also act trivially on the center ζG of G. Then （i） The extension →AutG＇G→AutG→AutG＇→1 is split.（ii）AutG＇G/AutG/ζG,ζGG≈Sp（2n,Z）×（GL（m,Z）×（Z）m）,（iii）Aut GζG,ζGG/InnG≈（Zk）2n＋（Z）2nm.

关键词： Generalized extraspecial Z-group symplectic group automorphism group

Automorphisms of Extensions of Q by a Direct Sum of Finitely Many Copies of Q

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Acta Mathematica Sinica,English Series 2019年第2期35卷 204-212页

作者： he guo liu Yu Lei WANG Ji Ping ZHANG Department of Mathematics Hubei University Department of Mathematics He'nan University of Technology The School of Mathematical Sciences Peking University

Let G be an extension of Q by a direct sum of r copies of Q.(1) If G is abelian, then G is a direct sum of r + 1 copies of Q and Aut G = GL(r + 1, Q);(2) If G is non-abelian, then G is a direct product of an extraspecial Q-group E and m copies of Q, where E/ζ E is a linear space over Q with dimension 2 n and m + 2 n = r. Furthermore, let Aut_G'G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and Aut_(G/ζG),_(ζG)G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G')G→ Aut G→ Aut G'→ 1 is split;(ii)Aut_(G')G/Aut_(G/ζG),_(ζG)G = Sp(2 n, Q) ×(GL(m, Q) Q^(m));(iii) Aut_(G/ζG),ζGG/Inn G= Q^(2 nm).

关键词： Extraspecial Q-group group extension symplectic group automorphism group

On the Unitriangular Groups over Rational Numbers Field

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Acta Mathematica Sinica,English Series 2022年第4期38卷 718-734页

作者： Rui GAO Jun LIAO he guo liu Xing Zhong XU School of Mathematics and Statistics Hubei UniversityWuhan 430062P.R.China

Let U(n,Q)be the group of all n×n(upper)unitriangular matrices over rational numbers field *** S be a subset of U(n,Q).In this paper,we prove that S is a subgroup of U(n,Q)if and only if the(i,j)-th entry S;satisfies some condition(see Theorem 3.5).Furthermore,we compute the upper central series and the lower central series for S,and obtain the condition that the upper central series and the lower central series of S coincide.

关键词： Nilpotent groups unitriangular groups central series

Finitely Generated Nilpotent Groups of Infinite Cyclic Commutator Subgroups

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Acta Mathematica Sinica,English Series 2020年第12期36卷 1315-1340页

作者： Jun LIAO he guo liu Xing Zhong XU Ji Ping ZHANG Hubei Key Laboratory of Applied Mathematics School of Mathematics and StatisticsHubei UniversityWuhan 430062P.R.China School of Mathematical Sciences Peking UniversityBeijing 100871P.R.China

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator *** the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group offinite rank of infinite cyclic center,we provide a decomposition of G as the product of a generalized extraspecial Z-group and its *** using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial Z-groups,we finally obtain the structure and invariants of the group G.

关键词： Nilpotent groups central extension isomorphic invariant