Let π be a set of primes. Isaacs established the π-theory of characters, which generalizes the theory of Brauer modular characters. Motivated by Isaacs's work, we introduce the definition of Mπ-groups and provide ...Let π be a set of primes. Isaacs established the π-theory of characters, which generalizes the theory of Brauer modular characters. Motivated by Isaacs's work, we introduce the definition of Mπ-groups and provide a characterization of Mπ-groups.展开更多
In this paper, We show that the simple K\-3-groups can be characterized by the orders of their maximal abelian subgroups. That is, we have Theorem Let G be a finite group and M a simple K \-3-group. Then ...In this paper, We show that the simple K\-3-groups can be characterized by the orders of their maximal abelian subgroups. That is, we have Theorem Let G be a finite group and M a simple K \-3-group. Then G is isomorphic to M if and only if the set of the orders of the maximal abelian subgoups of G is the same as that of M .展开更多
Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant ...Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite p-groups.展开更多
The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K...The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K2(R) =K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.展开更多
In order to answer a question motivated by constructing substitution boxes in block ciphers we will exhibit an infinite family of full-rank factorizations of elementary 2-groups into two factors having equal sizes.
We show that two irreducible operators on H are unitari1y equivalentif and only if W*(A B)’≌M2(C), and give an answer to the open question posedby J. B. Conway (Subnormal Operators, πPitman, Advanced Publishing Pro...We show that two irreducible operators on H are unitari1y equivalentif and only if W*(A B)’≌M2(C), and give an answer to the open question posedby J. B. Conway (Subnormal Operators, πPitman, Advanced Publishing Program,Boston, London, Melbourne, 1981) for irreducible operator. We also show that ifT, T1 and T2 are irreducible operators with T T1≌T T2, then T1≌T2. Finally,weshow that K0 (A(D))≌Z, giving a new result on the K0-group of Banach algebras.展开更多
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
Suppose that G is a finite p-group.If 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)-group.We useA0-group to denote an abelian grou...Suppose that G is a finite p-group.If 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)-group.We useA0-group to denote an abelian group.From 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 classified.Classifying A_(3)-groups is an old problem.In this paper,some general properties about A_(t)-groups are given.A_(3)-groups are completely classified up to isomorphism.Moreover,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.展开更多
Let G be a finite p-group.If the order of the derived subgroup of each proper subgroup of G divides pi,G is called a Di-group.In this paper,we give a characterization of all D1-groups.This is an answer to a question i...Let G be a finite p-group.If the order of the derived subgroup of each proper subgroup of G divides pi,G is called a Di-group.In this paper,we give a characterization of all D1-groups.This is an answer to a question introduced by Berkovich.展开更多
New groups: X-groups are constructed. These groups are easy to calculateand they reflect more general properties of rings than K0-groups. As their applications, some characteristics of group rings with torsion-free K0...New groups: X-groups are constructed. These groups are easy to calculateand they reflect more general properties of rings than K0-groups. As their applications, some characteristics of group rings with torsion-free K0-groups are obtained.展开更多
Letσ={σ_(i)|i∈I}be some partition of all primes P and G a finite group.A subgroup H of G is said to beσ-subnormal in G if there exists a subgroup chain H=H_(0)≤H_(1)≤・・・≤Hn=G such that either H_(i−1)is normal i...Letσ={σ_(i)|i∈I}be some partition of all primes P and G a finite group.A subgroup H of G is said to beσ-subnormal in G if there exists a subgroup chain H=H_(0)≤H_(1)≤・・・≤Hn=G such that either H_(i−1)is normal in Hi or Hi/(H_(i−1))Hi is a finiteσj-group for some j∈I for i=1,...,n.We call a finite group G a T_(σ)-group if everyσ-subnormal subgroup is normal in G.In this paper,we analyse the structure of the T_(σ)-groups and give some characterisations of the T_(σ)-groups.展开更多
A finite non-abelian group G is called metahamiltonian if every subgroup of G is either abelian or normal in G.If G is non-nilpotent,then the structure of G has been determined.If G is nilpotent,then the structure of ...A finite non-abelian group G is called metahamiltonian if every subgroup of G is either abelian or normal in G.If G is non-nilpotent,then the structure of G has been determined.If G is nilpotent,then the structure of G is determined by the structure of its Sylow subgroups.However,the classification of finite metahamiltonian p-groups is an unsolved problem.In this paper,finite metahamiltonian p-groups are completely classified up to isomorphism.展开更多
This paper is a survey on the recent work of the authors and their col-laborators on the Classification of Inductive Limit C*-algebras. Some examples are presented to explain several important ideas.
Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idemp...Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an "approximate inverse" of the Riesz decomposition theorem. In the paper, we give a more precise characterization of approximate invariants of strongly irreducible operators. The main result is: For any T ∈ L(H) with connected spectrum and ε > 0, there exists a strongly irreducible operator A, such that A - T < ε, V (A (A)) =~ N, K0(A (A)) ~= Z, and A (A)/rad A (A) is commutative, where A (A) denotes the commutant of A and rad A (A) denotes the Jacobson radical of A (A). The research is inspired by the recent similarity classification technique of Cowen-Douglas operators of Jiang Chunlan.展开更多
Let G be a finite abelian p-group,Г the maximal Z-order of Z[G]. We prove that the 2-primary torsion subgroups of K2Z[G]) and K2(Г)are isomorphic when p ≡ 3, 5, 7 (mod 8, and K2(Z[G])■zZ[1/p] is isomorphic to K2(...Let G be a finite abelian p-group,Г the maximal Z-order of Z[G]. We prove that the 2-primary torsion subgroups of K2Z[G]) and K2(Г)are isomorphic when p ≡ 3, 5, 7 (mod 8, and K2(Z[G])■zZ[1/p] is isomorphic to K2(Г)■zZ[1/p] when p = 2,3,5,7. As an application, we give the structure of K2(Z[G]) for G a cyclic p-group or an elementary abelian p-group.展开更多
Let G be a finite nonabelian group which has no abelian maximal subgroups and satisfies that any two non-commutative elements generate a maximal subgroup. Then G is isomorphic to the smallest Suzuki 2-group of order 64.
In this paper, we are concerned with the classification of operators on complex separable Hilbert spaces, in the unitary equivalence sense and the similarity sense, respectively. We show that two strongly irreducible ...In this paper, we are concerned with the classification of operators on complex separable Hilbert spaces, in the unitary equivalence sense and the similarity sense, respectively. We show that two strongly irreducible operators A and B are unitary equivalent if and only if W*(A+B)′≈M2(C), and two operators A and B in B1(Ω) are similar if and only if A′(AGB)/J≈M2(C). Moreover, we obtain V(H^∞(Ω,μ)≈N and Ko(H^∞(Ω,μ)≈Z by the technique of complex geometry, where Ω is a bounded connected open set in C, and μ is a completely non-reducing measure on Ω.展开更多
The famous von Neumann-Wold Theorem tells us that each analytic Toeplitz operator with n + 1-Blaschke factors is unitary to n + 1 copies of the unilateral shift on the Hardy space. It is obvious that the von Neumann-W...The famous von Neumann-Wold Theorem tells us that each analytic Toeplitz operator with n + 1-Blaschke factors is unitary to n + 1 copies of the unilateral shift on the Hardy space. It is obvious that the von Neumann-Wold Theorem does not hold in the Bergman space. In this paper, using the basis constructed by Michael and Zhu on the Bergman space we prove that each analytic Toeplitz operator Mb(z) is similar to n + 1 copies of the Bergman shift if and only if B(z) is an n + 1-Blaschke product. Prom the above theorem, we characterize the similarity invariant of some analytic Toeplitz operators by using K0-group term.展开更多
For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field an...For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field and if n = 4,8,12 is a positive integer having a square factor then Gn(F) is not a subgroup of K2(F),and then by using the results of Manin,Grauert,Samuel and Li on Mordell conjecture theorem for function fields,a similar result is established for function fields over an algebraically closed field.展开更多
基金Supported by NSF of China(1117116911071155)the B.S.Foundation of Shandong Province(BS2012SF003)
文摘Let π be a set of primes. Isaacs established the π-theory of characters, which generalizes the theory of Brauer modular characters. Motivated by Isaacs's work, we introduce the definition of Mπ-groups and provide a characterization of Mπ-groups.
文摘In this paper, We show that the simple K\-3-groups can be characterized by the orders of their maximal abelian subgroups. That is, we have Theorem Let G be a finite group and M a simple K \-3-group. Then G is isomorphic to M if and only if the set of the orders of the maximal abelian subgoups of G is the same as that of M .
文摘Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite p-groups.
文摘The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K2(R) =K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.
文摘In order to answer a question motivated by constructing substitution boxes in block ciphers we will exhibit an infinite family of full-rank factorizations of elementary 2-groups into two factors having equal sizes.
基金The 973 Project of China and the NNSF (Grant No. 19631070) of China.
文摘We show that two irreducible operators on H are unitari1y equivalentif and only if W*(A B)’≌M2(C), and give an answer to the open question posedby J. B. Conway (Subnormal Operators, πPitman, Advanced Publishing Program,Boston, London, Melbourne, 1981) for irreducible operator. We also show that ifT, T1 and T2 are irreducible operators with T T1≌T T2, then T1≌T2. Finally,weshow that K0 (A(D))≌Z, giving a new result on the K0-group of Banach algebras.
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This work was supported by NSFC(Nos.11371232,11471198)by NSF of Shanxi Province(No.2013011001).
文摘Suppose that G is a finite p-group.If 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)-group.We useA0-group to denote an abelian group.From 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 classified.Classifying A_(3)-groups is an old problem.In this paper,some general properties about A_(t)-groups are given.A_(3)-groups are completely classified up to isomorphism.Moreover,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.
基金supported by National Natural Science Foundation of China (Grant Nos.10571128,10871032)Natural Science Foundation of Jiangsu Province (Grant No.BK2008156)Suzhou City Senior Talent Supporting Project
文摘Let G be a finite p-group.If the order of the derived subgroup of each proper subgroup of G divides pi,G is called a Di-group.In this paper,we give a characterization of all D1-groups.This is an answer to a question introduced by Berkovich.
基金Project supported by the National Natural Science Foundation of China.
文摘New groups: X-groups are constructed. These groups are easy to calculateand they reflect more general properties of rings than K0-groups. As their applications, some characteristics of group rings with torsion-free K0-groups are obtained.
文摘Letσ={σ_(i)|i∈I}be some partition of all primes P and G a finite group.A subgroup H of G is said to beσ-subnormal in G if there exists a subgroup chain H=H_(0)≤H_(1)≤・・・≤Hn=G such that either H_(i−1)is normal in Hi or Hi/(H_(i−1))Hi is a finiteσj-group for some j∈I for i=1,...,n.We call a finite group G a T_(σ)-group if everyσ-subnormal subgroup is normal in G.In this paper,we analyse the structure of the T_(σ)-groups and give some characterisations of the T_(σ)-groups.
基金This work was supported by NSFC(Nos.11971280,11771258).
文摘A finite non-abelian group G is called metahamiltonian if every subgroup of G is either abelian or normal in G.If G is non-nilpotent,then the structure of G has been determined.If G is nilpotent,then the structure of G is determined by the structure of its Sylow subgroups.However,the classification of finite metahamiltonian p-groups is an unsolved problem.In this paper,finite metahamiltonian p-groups are completely classified up to isomorphism.
基金Both authors are supported by NSF grant DMS9970840 This material is also based uponwork supported by,the U.S. Army Research Office under grant number DAADl9-00-1-0152 for both authors.
文摘This paper is a survey on the recent work of the authors and their col-laborators on the Classification of Inductive Limit C*-algebras. Some examples are presented to explain several important ideas.
基金supported by the National Natural Science Foundation of China (Grant No. 10571041)Hebei Provincial Natural Science Foundation (Grant No. A2005000006)
文摘Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an "approximate inverse" of the Riesz decomposition theorem. In the paper, we give a more precise characterization of approximate invariants of strongly irreducible operators. The main result is: For any T ∈ L(H) with connected spectrum and ε > 0, there exists a strongly irreducible operator A, such that A - T < ε, V (A (A)) =~ N, K0(A (A)) ~= Z, and A (A)/rad A (A) is commutative, where A (A) denotes the commutant of A and rad A (A) denotes the Jacobson radical of A (A). The research is inspired by the recent similarity classification technique of Cowen-Douglas operators of Jiang Chunlan.
基金This work was supported by the National Natural Science Foundation for Young Scientists of China (Grant No. 11401412the National Natural Science Foundation of China (Grant No. 11771422)the Scientific Research Foundation of the Higher Education Institutions of Jiangsu Province, China (Grant No. 18KJB110025).
文摘Let G be a finite abelian p-group,Г the maximal Z-order of Z[G]. We prove that the 2-primary torsion subgroups of K2Z[G]) and K2(Г)are isomorphic when p ≡ 3, 5, 7 (mod 8, and K2(Z[G])■zZ[1/p] is isomorphic to K2(Г)■zZ[1/p] when p = 2,3,5,7. As an application, we give the structure of K2(Z[G]) for G a cyclic p-group or an elementary abelian p-group.
基金NSFC (No.10671114)NSF of Shanxi Province (No.20051007)the Returned Overseas(student) Fund of Shanxi province (No.[2007]13-56)
文摘Let G be a finite nonabelian group which has no abelian maximal subgroups and satisfies that any two non-commutative elements generate a maximal subgroup. Then G is isomorphic to the smallest Suzuki 2-group of order 64.
基金the 973 Project of China and the National Natural Science Foundation of China(Grant No.19631070)
文摘In this paper, we are concerned with the classification of operators on complex separable Hilbert spaces, in the unitary equivalence sense and the similarity sense, respectively. We show that two strongly irreducible operators A and B are unitary equivalent if and only if W*(A+B)′≈M2(C), and two operators A and B in B1(Ω) are similar if and only if A′(AGB)/J≈M2(C). Moreover, we obtain V(H^∞(Ω,μ)≈N and Ko(H^∞(Ω,μ)≈Z by the technique of complex geometry, where Ω is a bounded connected open set in C, and μ is a completely non-reducing measure on Ω.
基金the National Natural Science Foundation of China (Grant No. 10571041)
文摘The famous von Neumann-Wold Theorem tells us that each analytic Toeplitz operator with n + 1-Blaschke factors is unitary to n + 1 copies of the unilateral shift on the Hardy space. It is obvious that the von Neumann-Wold Theorem does not hold in the Bergman space. In this paper, using the basis constructed by Michael and Zhu on the Bergman space we prove that each analytic Toeplitz operator Mb(z) is similar to n + 1 copies of the Bergman shift if and only if B(z) is an n + 1-Blaschke product. Prom the above theorem, we characterize the similarity invariant of some analytic Toeplitz operators by using K0-group term.
基金supported by the National Natural Science Foundation of China (Grant No.10371061)
文摘For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field and if n = 4,8,12 is a positive integer having a square factor then Gn(F) is not a subgroup of K2(F),and then by using the results of Manin,Grauert,Samuel and Li on Mordell conjecture theorem for function fields,a similar result is established for function fields over an algebraically closed field.