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
Let G be a p-group (p odd prime) and let X = Cay(G, S) be a 4-valent connected Cayley graph. It is shown that if G has nilpotent class 2, then the automorphism group Ant(X) of X is isomorphic to the semidirect product...Let G be a p-group (p odd prime) and let X = Cay(G, S) be a 4-valent connected Cayley graph. It is shown that if G has nilpotent class 2, then the automorphism group Ant(X) of X is isomorphic to the semidirect product GR x Ant(G,S), where GR is the right regular representation of G and Aut(G,S) is the subgroup of the automorphism group Aut(G) of G which fixes S setwise. However the result is not true if G has nilpotent class 3 and this paper provides a counterexample.展开更多
We prove that certain 1-relator groups have Property E. Using this fact, we characterize all conjugacy separable 1-relator groups of the form a,b;(a-αbβaαbγ)t , t 1, having residually finite outer automorphism gro...We prove that certain 1-relator groups have Property E. Using this fact, we characterize all conjugacy separable 1-relator groups of the form a,b;(a-αbβaαbγ)t , t 1, having residually finite outer automorphism groups.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
In this paper, we determine the order of automorphism group of p-groups in the third family ( Φ 3) and the fourth family ( Φ 4) in [1], whose order is p^6(p≥3). Here p denotes an odd prime.
We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. We introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automor...We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. We introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra(i.e., the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients,cluster algebras with universal geometric coefficients, and cluster algebras from surfaces(except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.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
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
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
Let W = {ω1,ω2,…,ωn-1} be a minimal generating transposition set of Sn. In this paper it was shown that V = （0, p）W = {（0, p）ω1, （0, p）ω2…, （0,p）ωn-1} is a minimal generating set of alternating group A...Let W = {ω1,ω2,…,ωn-1} be a minimal generating transposition set of Sn. In this paper it was shown that V = （0, p）W = {（0, p）ω1, （0, p）ω2…, （0,p）ωn-1} is a minimal generating set of alternating group An＋1, where p ∈{1, 2, …, n}. And then we investigated the automorphism groups of Cayley graphs on alternating groups with these generating sets which contain alternating group graph AGn.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
It is proved that for a complex minimal smooth projective surface S of general type, its abelian automorphism group is of order≤36k2s+24, provided x(os)≥8, where Ks is the canonical divisor of S, and X(Os) the Euler...It is proved that for a complex minimal smooth projective surface S of general type, its abelian automorphism group is of order≤36k2s+24, provided x(os)≥8, where Ks is the canonical divisor of S, and X(Os) the Euler characteristic of the structure sheaf of S.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
In this paper,the authors determine maximal connected automorphism group of the Lie transformation group T（D（VN,F））,which acting on the normal Siegel domain D（VN,F）is simple and transitive,and prove that the max...In this paper,the authors determine maximal connected automorphism group of the Lie transformation group T（D（VN,F））,which acting on the normal Siegel domain D（VN,F）is simple and transitive,and prove that the maximal connected automorphism group of T（D（VN,F））is its maximal connected inner automorphism group.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the National Natural Science Foundation of China (No.10071002) andCom2MaC-KOSEF.
文摘Let G be a p-group (p odd prime) and let X = Cay(G, S) be a 4-valent connected Cayley graph. It is shown that if G has nilpotent class 2, then the automorphism group Ant(X) of X is isomorphic to the semidirect product GR x Ant(G,S), where GR is the right regular representation of G and Aut(G,S) is the subgroup of the automorphism group Aut(G) of G which fixes S setwise. However the result is not true if G has nilpotent class 3 and this paper provides a counterexample.
基金supported by Korean Research Foundation funded by the Korean Government (Grant No. KRF-2007-313-C00015)The second author was supported by Natural Science and Engineering Research Council of Canada (Grant No. A-4064)
文摘We prove that certain 1-relator groups have Property E. Using this fact, we characterize all conjugacy separable 1-relator groups of the form a,b;(a-αbβaαbγ)t , t 1, having residually finite outer automorphism groups.
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).
文摘In this paper, we determine the order of automorphism group of p-groups in the third family ( Φ 3) and the fourth family ( Φ 4) in [1], whose order is p^6(p≥3). Here p denotes an odd prime.
基金supported by National Natural Science Foundation of China(Grant No.11131001)
文摘We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. We introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra(i.e., the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients,cluster algebras with universal geometric coefficients, and cluster algebras from surfaces(except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.
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.
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,∞}.
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This work is supported by National Natural Science Foundation of China （10671081） Scientific and Techno- logical Project of Hubei Province （2006AA412C27） Science Foundation of Three Gorges University （604401）.
文摘Let W = {ω1,ω2,…,ωn-1} be a minimal generating transposition set of Sn. In this paper it was shown that V = （0, p）W = {（0, p）ω1, （0, p）ω2…, （0,p）ωn-1} is a minimal generating set of alternating group An＋1, where p ∈{1, 2, …, n}. And then we investigated the automorphism groups of Cayley graphs on alternating groups with these generating sets which contain alternating group graph AGn.
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.
文摘It is proved that for a complex minimal smooth projective surface S of general type, its abelian automorphism group is of order≤36k2s+24, provided x(os)≥8, where Ks is the canonical divisor of S, and X(Os) the Euler characteristic of the structure sheaf of S.
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).
基金Supported by the National Science Foundation of China（11047030） Supported by the Natural Science Foundation of Henan Provincial Education Department（2010B11003） Supported by the Natural Science Foundation of Henan University（2009YBZR025）
文摘In this paper,the authors determine maximal connected automorphism group of the Lie transformation group T（D（VN,F））,which acting on the normal Siegel domain D（VN,F）is simple and transitive,and prove that the maximal connected automorphism group of T（D（VN,F））is its maximal connected inner automorphism group.
