Let p be a prime and F_p be a finite field of p elements.Let 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 ...Let p be a prime and F_p be a finite field of p elements.Let 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≥1.Then V(F_(p)G)≌V(F_(p)H)if and only if G≌H.Balogh and Bovdi only solved the isomorphism problem when p is odd.In this paper,the case p=2 is determined.展开更多
Let U(n,Q)be the group of all n×n(upper)unitriangular matrices over rational numbers field Q.Let 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;sat...Let U(n,Q)be the group of all n×n(upper)unitriangular matrices over rational numbers field Q.Let 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.展开更多
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, w...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.展开更多
Let p be an odd prime,and let k be a nonzero nature number.Suppose 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’.Th...Let p be an odd prime,and let k be a nonzero nature number.Suppose 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 andζG.Let|E|=p(2n+1)k and|ζG|=p(m+1)k.Suppose that the exponents of E andζG are pk+l and pk+r,respectively,where 0≤l,r≤k.Let 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).展开更多
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...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 the condition on surjectivity, we prove that C;（α;） and G/[G, α;] are both abelian-by-finite.展开更多
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...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 unique components are isomorphic to the exotic fusion systems over 7+^1+2 found by Ruiz and Viruel.展开更多
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 extraspec...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).展开更多
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 × · · &#...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,and |P'|≤p,m≥2. The purpose of this paper is to prove that P is resistant.展开更多
Let G and N be arbitrary groups. In this paper, we construct an associated group extension εx of N/Z（N） by G for any group homomorphism X ： G → Out N, and prove that X can be lifted to a group action, that is, a ...Let G and N be arbitrary groups. In this paper, we construct an associated group extension εx of N/Z（N） by G for any group homomorphism X ： G → Out N, and prove that X can be lifted to a group action, that is, a group homomorphism from G to Aut N, if and only if the extension εx splits. Furthermore we obtain an explicit description for all such lifting homomorphisms and the number of its conjugacy classes, and give an application of the lifting technique of outer actions to the theory of group extensions.展开更多
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 subgroup.Using the structure and invariant of the...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 subgroup.Using 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 center.By 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.展开更多
基金National Natural Science Foundation of China(Grant No.12171142)。
文摘Let p be a prime and F_p be a finite field of p elements.Let 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≥1.Then V(F_(p)G)≌V(F_(p)H)if and only if G≌H.Balogh and Bovdi only solved the isomorphism problem when p is odd.In this paper,the case p=2 is determined.
基金National Natural Science Foundation of China(Grant Nos.12171142,11971155,12071117)。
文摘Let U(n,Q)be the group of all n×n(upper)unitriangular matrices over rational numbers field Q.Let 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.
基金Supported by NSFC(Grant Nos.11771129 and 11601121)Henan Provincial Natural Science Foundation of China(Grant No.162300410066)Program for Innovation Talents of Science and Technology of Henan University of Technology(Grant No.11CXRC19)
文摘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.
基金Supported by NSFC(Grant Nos.11601121,11771129)Natural Science Foundation of He’nan Province of China(Grant No.162300410066)。
文摘Let p be an odd prime,and let k be a nonzero nature number.Suppose 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 andζG.Let|E|=p(2n+1)k and|ζG|=p(m+1)k.Suppose that the exponents of E andζG are pk+l and pk+r,respectively,where 0≤l,r≤k.Let 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).
基金Supported by National Natural Science Foundation of China(Grant No.11371124)Youth Foundation of Hebei Educational Committee(Grant Nos.QN2016184 and F2015402033)Graduate Education Teaching Reform Foundation of Hebei University of Engineering(Grant No.161290140004)
文摘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 the condition on surjectivity, we prove that C;（α;） and G/[G, α;] are both abelian-by-finite.
基金Supported by NSFC(Grant Nos.11131001,11371124 and 11401186)
文摘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 unique components are isomorphic to the exotic fusion systems over 7+^1+2 found by Ruiz and Viruel.
基金Supported by NSFC(Grant Nos.11771129 and 11601121)He'nan Provincial Natural Science Foundation of China(Grant No.162300410066)
文摘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).
基金Supported by NSFC(Grant Nos.11371154,11301150 and 11601121)Natural Science Foundation of Henan Province of China(Grant Nos.142300410134,162300410066)
文摘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,and |P'|≤p,m≥2. The purpose of this paper is to prove that P is resistant.
基金Supported by National Natural Science Foundation of China (Grant No. 10671058)
文摘Let G and N be arbitrary groups. In this paper, we construct an associated group extension εx of N/Z（N） by G for any group homomorphism X ： G → Out N, and prove that X can be lifted to a group action, that is, a group homomorphism from G to Aut N, if and only if the extension εx splits. Furthermore we obtain an explicit description for all such lifting homomorphisms and the number of its conjugacy classes, and give an application of the lifting technique of outer actions to the theory of group extensions.
基金Supported by NSFC(Grant Nos.11631001,11771129,11971155 and 12071117)。
文摘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 subgroup.Using 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 center.By 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.
