The orders of automorphism groups of the groups of order p^6 in the twelve family Ф12 axe produced, where p is an odd prime. Every group is analysed by utilizing the properties of metabelian, regularity and p-commuta...The orders of automorphism groups of the groups of order p^6 in the twelve family Ф12 axe produced, where p is an odd prime. Every group is analysed by utilizing the properties of metabelian, regularity and p-commutativity of finite p-groups, and the structure of the generators of its automorphism groups is obtained. Then the orders of automorphism groups are determined through some properties of equivalence in number theory.展开更多
The spectrum of a finite group is the set of its element orders,and two groups are said to be isospectral if they have the same spectra.A finite group G is said to be recognizable by spectrum,if every finite group iso...The spectrum of a finite group is the set of its element orders,and two groups are said to be isospectral if they have the same spectra.A finite group G is said to be recognizable by spectrum,if every finite group isospectral with G is isomorphic to G.We prove that if S is one of the sporadic simple groups M^(c)L,M_(12),M_(22),He,Suz and O'N,then Aut(S)is recognizable by spectrum.This finishes the proof of the recognizability by spectrum of the automorphism groups of all sporadic simple groups,except J_(2).展开更多
For G a finite group,π_e(G)denotes the set of orders of elements in G.If Ω is a subset of the set of natural numbers,h(Ω)stands for the number of isomorphism classes of finite groups with the same set Ω of element...For G a finite group,π_e(G)denotes the set of orders of elements in G.If Ω is a subset of the set of natural numbers,h(Ω)stands for the number of isomorphism classes of finite groups with the same set Ω of element orders.We say that G is k-distinguishable if h(π_(G))=k<∞,otherwise G is called non-distinguishable.Usually,a 1-distinguishable group is called a characterizable group.It is shown that if M is a sporadic simple group different from M_(12),M_(22),J_2,He,Suz,M^cL and O'N, then Aut(M)is charaeterizable by its dement orders.It is also proved that if M is isomorphic to M_(12),M_(22),He,Suz or O'N,then h(π_e(Aut(M)))∈{1,∞}.展开更多
In this paper, a finite group G with IAut(G) : P(G)I ^- p or pq is determined, where P(G) is the power automorphism group of G, and p, q are distinct primes. Especially, we prove that a finite group G satisfies...In this paper, a finite group G with IAut(G) : P(G)I ^- p or pq is determined, where P(G) is the power automorphism group of G, and p, q are distinct primes. Especially, we prove that a finite group G satisfies |Aut(G) : P(G)|= pq if and only if Aut(G)/P(G) ≌S3. Also, some other classes of finite groups are investigated and classified, which are necessary for the proof of our main results.展开更多
This paper is devoted to a study of the automorphism groups of three series of finite-dimensional special odd Hamiltonian superalgebras g over a field of prime characteristic. Our aim is to characterize the connection...This paper is devoted to a study of the automorphism groups of three series of finite-dimensional special odd Hamiltonian superalgebras g over a field of prime characteristic. Our aim is to characterize the connections between the automorphism groups of g and the automorphism groups of the corresponding underlying superalgebras. Precisely speaking, we embed the former into the later. Moreover, we determine the images of the normal series of the automorphism groups and homogeneous automorphism groups of g under the embedded mapping.展开更多
The authors study and classify finite groups with their automorphism groups having orders 4pq2, where p and q axe primes such that 2 〈 p 〈 q. In 2013 the first and the third authors classified the nilpotent groups o...The authors study and classify finite groups with their automorphism groups having orders 4pq2, where p and q axe primes such that 2 〈 p 〈 q. In 2013 the first and the third authors classified the nilpotent groups of this kind; here the authors give the classification of those finite non-nilpotent groups with their automorphism groups having orders 4pq2.展开更多
We study the relations between two groups related to cluster automorphism groups which are defined by Assem,Schiffler and Shamchenko.We establish the relation-ships among(strict)direct cluster automorphism groups and ...We study the relations between two groups related to cluster automorphism groups which are defined by Assem,Schiffler and Shamchenko.We establish the relation-ships among(strict)direct cluster automorphism groups and those groups consisting of periodicities of labeled seeds and exchange matrices,respectively,in the language of short exact sequences.As an application,we characterize automorphism-finite cluster algebras in the cases of bipartite seeds or finite mutation type.Finally,we study the relation between the group Aut(A)for a cluster algebra A and the group AutMn(S)for a mutation group Mn and a labeled mutation class S,and we give a negative answer via counter-examples to King and Pressland's problem.展开更多
Let Sn denote the symmetric group of degree n with n 〉 3, S = {cn = (1 2…n), cn^-1, (1 2)} and Fn=Cay(Sn, S) be the Cayley graph on Sn with respect to S. In this paper, we show that Fn (n 〉 13) is a normal ...Let Sn denote the symmetric group of degree n with n 〉 3, S = {cn = (1 2…n), cn^-1, (1 2)} and Fn=Cay(Sn, S) be the Cayley graph on Sn with respect to S. In this paper, we show that Fn (n 〉 13) is a normal Cayley graph, and that the full automorphism group of Fn is equal to Aut(Гn) = R(Sn) (Inn(C))≌- Sn×Z2, where R(Sn) is the right regular representation of Sn,φ = (1 2)(3 n)(4 n-1)(5 n-2)... (∈ Sn), and Inn(C) is the inner isomorphism of Sn induced by φ .展开更多
In this paper,the automorphism group of G is determined,where G is a 4 × 4 upper unitriangular matrix group over Z.Let K be the subgroup of AutG consisting of all elements of AutG which act trivially on G/G,G /ζ...In this paper,the automorphism group of G is determined,where G is a 4 × 4 upper unitriangular matrix group over Z.Let K be the subgroup of AutG consisting of all elements of AutG which act trivially on G/G,G /ζG and ζG,then (i) InnG ■ K ■ AutG;(ii) AutG/K≌=G_1×D_8×Z_2,where G_1=(a,b,c|a^4=b^2=c^2=1,a^b=a^(-1),[a,c]= [b,c]=1 ;(iii) K/Inn G≌=Z×Z×Z.展开更多
Suppose F is a field of characteristic not 2 and F* its multiplicative group. Let T*n(F) be the multiplicative group of invertible upper triangular n x n matrices over F and STn(F) its subgroup {(aij) E T*n(F)aii = 1,...Suppose F is a field of characteristic not 2 and F* its multiplicative group. Let T*n(F) be the multiplicative group of invertible upper triangular n x n matrices over F and STn(F) its subgroup {(aij) E T*n(F)aii = 1, i}. This paper proves that f: T*n(F) → T*n(F) is a group automorphism if and only if there exist a matrix Q in T*n(F) and a field automorphism rs of F such that either where A = ((aij)), A-T is the transpose inverse of A, J = Ei n+1-i, and : i= 1T*n(F) → F* is a homomorphism which satisfies {(xIn)(x)x F*} = F* and {x F*(xIn)(x) = 1} = {1}. Simultaneously, they also determine the automorphisms of STn(F).展开更多
We know that for a code C,it‘s very important to find out the Automorphism groupAutC of C.However,it is very difficult to seek entire AutC.In this paper,using the G.I of matrices over a finite field,we give several m...We know that for a code C,it‘s very important to find out the Automorphism groupAutC of C.However,it is very difficult to seek entire AutC.In this paper,using the G.I of matrices over a finite field,we give several methods to judge whether a permutation σ∈S_n.(Symmetric group) belongs to AutC or not.They are helpful for the purpose to ex-展开更多
we have discussed structures of Abelian group G by order |A(G) |of automoorphism group and have obtained all types of finite Abelian grooup G when the order of A(G) equals 27pq(p,q are odd primmes).
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
Fixed point subalgebras of finite dimensional factor algebras of algebras of polynomials in n indeterminates over the finite field F2 (with respect to all F2-algebra automorphisms) are fully described.
In the field of several complex variables, the Greene-Krantz Conjecture, whose consequences would be far reaching, has yet to be proven. The conjecture is as follows: Let D be a smoothly bounded domain in Cn. Suppose ...In the field of several complex variables, the Greene-Krantz Conjecture, whose consequences would be far reaching, has yet to be proven. The conjecture is as follows: Let D be a smoothly bounded domain in Cn. Suppose there exists {gj} ? Aut(D) such that {gj(z)} accumulates at a boundary point p ∈ ?D for some z ∈ D. Then ?D is of finite type at p. In this paper, we prove the following result, yielding further evidence to the probable veracity of this important conjecture: Let D be a bounded convex domain in C^2 with C^2 boundary.Suppose that there is a sequence {gj} ? Aut(D) such that {gj(z)} accumulates at a boundary point for some point z ∈ D. Then if p ∈ ?D is such an orbit accumulation point, ?D contains no non-trivial analytic variety passing through p.展开更多
基金The Science Research Foundation of Chongqing Municipal Education Commission of China(KJ050611)
文摘The orders of automorphism groups of the groups of order p^6 in the twelve family Ф12 axe produced, where p is an odd prime. Every group is analysed by utilizing the properties of metabelian, regularity and p-commutativity of finite p-groups, and the structure of the generators of its automorphism groups is obtained. Then the orders of automorphism groups are determined through some properties of equivalence in number theory.
基金This work is supported by Russian Science Foundation(Project No.14-21-00065).
文摘The spectrum of a finite group is the set of its element orders,and two groups are said to be isospectral if they have the same spectra.A finite group G is said to be recognizable by spectrum,if every finite group isospectral with G is isomorphic to G.We prove that if S is one of the sporadic simple groups M^(c)L,M_(12),M_(22),He,Suz and O'N,then Aut(S)is recognizable by spectrum.This finishes the proof of the recognizability by spectrum of the automorphism groups of all sporadic simple groups,except J_(2).
基金This work has been partially sopported by the Research Institute for Fundamental Sciences Tabriz,Iran
文摘For G a finite group,π_e(G)denotes the set of orders of elements in G.If Ω is a subset of the set of natural numbers,h(Ω)stands for the number of isomorphism classes of finite groups with the same set Ω of element orders.We say that G is k-distinguishable if h(π_(G))=k<∞,otherwise G is called non-distinguishable.Usually,a 1-distinguishable group is called a characterizable group.It is shown that if M is a sporadic simple group different from M_(12),M_(22),J_2,He,Suz,M^cL and O'N, then Aut(M)is charaeterizable by its dement orders.It is also proved that if M is isomorphic to M_(12),M_(22),He,Suz or O'N,then h(π_e(Aut(M)))∈{1,∞}.
基金supported by National Natural Science Foundation of China (Grant No. 10771132)SGRC (Grant No. GZ310)+1 种基金Shanghai Leading Academic Discipline Project (Grant No. J50101)Science Technology Foundation of Shanxi Province for Colleges (Grant No. 20081022)
文摘In this paper, a finite group G with IAut(G) : P(G)I ^- p or pq is determined, where P(G) is the power automorphism group of G, and p, q are distinct primes. Especially, we prove that a finite group G satisfies |Aut(G) : P(G)|= pq if and only if Aut(G)/P(G) ≌S3. Also, some other classes of finite groups are investigated and classified, which are necessary for the proof of our main results.
文摘This paper is devoted to a study of the automorphism groups of three series of finite-dimensional special odd Hamiltonian superalgebras g over a field of prime characteristic. Our aim is to characterize the connections between the automorphism groups of g and the automorphism groups of the corresponding underlying superalgebras. Precisely speaking, we embed the former into the later. Moreover, we determine the images of the normal series of the automorphism groups and homogeneous automorphism groups of g under the embedded mapping.
基金Supported by National Natural Science Foundation of China (Nos. 11671324, 11471266, 11501071).
文摘The authors study and classify finite groups with their automorphism groups having orders 4pq2, where p and q axe primes such that 2 〈 p 〈 q. In 2013 the first and the third authors classified the nilpotent groups of this kind; here the authors give the classification of those finite non-nilpotent groups with their automorphism groups having orders 4pq2.
文摘We study the relations between two groups related to cluster automorphism groups which are defined by Assem,Schiffler and Shamchenko.We establish the relation-ships among(strict)direct cluster automorphism groups and those groups consisting of periodicities of labeled seeds and exchange matrices,respectively,in the language of short exact sequences.As an application,we characterize automorphism-finite cluster algebras in the cases of bipartite seeds or finite mutation type.Finally,we study the relation between the group Aut(A)for a cluster algebra A and the group AutMn(S)for a mutation group Mn and a labeled mutation class S,and we give a negative answer via counter-examples to King and Pressland's problem.
基金This work is supported by the National Natural Science Foundation of China (Grant Nos. 11671344 and 11531011).
文摘Let Sn denote the symmetric group of degree n with n 〉 3, S = {cn = (1 2…n), cn^-1, (1 2)} and Fn=Cay(Sn, S) be the Cayley graph on Sn with respect to S. In this paper, we show that Fn (n 〉 13) is a normal Cayley graph, and that the full automorphism group of Fn is equal to Aut(Гn) = R(Sn) (Inn(C))≌- Sn×Z2, where R(Sn) is the right regular representation of Sn,φ = (1 2)(3 n)(4 n-1)(5 n-2)... (∈ Sn), and Inn(C) is the inner isomorphism of Sn induced by φ .
基金Supported by the Tianyuan Fund for Mathematics of NSFC(11126273)Supported by the NSF of Henan Educational Committee(2011B110011)Supported by the Doctor Foundation of Henan University of Technology(2009BS029)
文摘In this paper,the automorphism group of G is determined,where G is a 4 × 4 upper unitriangular matrix group over Z.Let K be the subgroup of AutG consisting of all elements of AutG which act trivially on G/G,G /ζG and ζG,then (i) InnG ■ K ■ AutG;(ii) AutG/K≌=G_1×D_8×Z_2,where G_1=(a,b,c|a^4=b^2=c^2=1,a^b=a^(-1),[a,c]= [b,c]=1 ;(iii) K/Inn G≌=Z×Z×Z.
基金This work is supported by NSF of China NSF of Heilongjiang province
文摘Suppose F is a field of characteristic not 2 and F* its multiplicative group. Let T*n(F) be the multiplicative group of invertible upper triangular n x n matrices over F and STn(F) its subgroup {(aij) E T*n(F)aii = 1, i}. This paper proves that f: T*n(F) → T*n(F) is a group automorphism if and only if there exist a matrix Q in T*n(F) and a field automorphism rs of F such that either where A = ((aij)), A-T is the transpose inverse of A, J = Ei n+1-i, and : i= 1T*n(F) → F* is a homomorphism which satisfies {(xIn)(x)x F*} = F* and {x F*(xIn)(x) = 1} = {1}. Simultaneously, they also determine the automorphisms of STn(F).
文摘We know that for a code C,it‘s very important to find out the Automorphism groupAutC of C.However,it is very difficult to seek entire AutC.In this paper,using the G.I of matrices over a finite field,we give several methods to judge whether a permutation σ∈S_n.(Symmetric group) belongs to AutC or not.They are helpful for the purpose to ex-
文摘we have discussed structures of Abelian group G by order |A(G) |of automoorphism group and have obtained all types of finite Abelian grooup G when the order of A(G) equals 27pq(p,q are odd primmes).
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the subsidization of the GA CR,grantNo.201/09/0981.
文摘Fixed point subalgebras of finite dimensional factor algebras of algebras of polynomials in n indeterminates over the finite field F2 (with respect to all F2-algebra automorphisms) are fully described.
文摘In the field of several complex variables, the Greene-Krantz Conjecture, whose consequences would be far reaching, has yet to be proven. The conjecture is as follows: Let D be a smoothly bounded domain in Cn. Suppose there exists {gj} ? Aut(D) such that {gj(z)} accumulates at a boundary point p ∈ ?D for some z ∈ D. Then ?D is of finite type at p. In this paper, we prove the following result, yielding further evidence to the probable veracity of this important conjecture: Let D be a bounded convex domain in C^2 with C^2 boundary.Suppose that there is a sequence {gj} ? Aut(D) such that {gj(z)} accumulates at a boundary point for some point z ∈ D. Then if p ∈ ?D is such an orbit accumulation point, ?D contains no non-trivial analytic variety passing through p.