In this paper, we investigate the algebraic structure of certain 2-generator groups of permutations of the integers. The groups fall into two infinite classes: one class terminates with the quaternion group and the ot...In this paper, we investigate the algebraic structure of certain 2-generator groups of permutations of the integers. The groups fall into two infinite classes: one class terminates with the quaternion group and the other class terminates with the Klein-four group. We show that all the groups are finitely presented and we determine minimal presentations in each case. Finally, we determine the order of each group.展开更多
Representation theory is concerned with the ways of explaining or visualizing a group as a group of matrices. In this paper, we extend the permutation pattern of to a two-line notation. We consider the representations...Representation theory is concerned with the ways of explaining or visualizing a group as a group of matrices. In this paper, we extend the permutation pattern of to a two-line notation. We consider the representations of this non-deranged permutation group(p ≥ 5 and p a prime). Also we reveal some interesting properties and results of the character of where .展开更多
Let n be a positive integer. A permutation a of the symmetric group of permutations of is called a derangement if for each . Suppose that x and y are two arbitrary permutations of . We say that...Let n be a positive integer. A permutation a of the symmetric group of permutations of is called a derangement if for each . Suppose that x and y are two arbitrary permutations of . We say that a permutation a is a double derangement with respect to x and y if and for each . In this paper, we give an explicit formula for , the number of double derangements with respect to x and y. Let and let and be two subsets of with and . Suppose that denotes the number of derangements x such that . As the main result, we show that if and z is a permutation such that for and for , then where .展开更多
The application of χ state are investigated in remote state preparation （RSP）. By constructing useful measurement bases with the aid of Hurwitz matrix equation, we propose several RSP schemes of arbitrary two- and ...The application of χ state are investigated in remote state preparation （RSP）. By constructing useful measurement bases with the aid of Hurwitz matrix equation, we propose several RSP schemes of arbitrary two- and three-qubit states via the χ state as the entangled resource. It is shown that the original state can be successfully prepared with the probability 100% and 50% for real coefficients and complex coefficients, respectively. For the latter case, the special ensembles with unit success probability are discussed by the permutation group. It is worth mentioning that the novel measurement bases have no restrictions on the coefficients of the prepared state, which means that the proposed schemes are more applicable.展开更多
Some classical results about linear representations of a finite group G have been also proved for representations of G on non-abelian groups （G-groups）. In this paper we establish a decomposition theorem for irreduc...Some classical results about linear representations of a finite group G have been also proved for representations of G on non-abelian groups （G-groups）. In this paper we establish a decomposition theorem for irreducible G-groups which expresses a suitable irreducible G-group as a tensor product of two projective G-groups in a similar way to the celebrated theorem of Clifford for linear representations. Moreover, we study the non-abelian minimal normal subgroups of G in which this decomposition is possible.展开更多
Let G be a finite group. Fix a prime divisor p of IGI and a Sylow p-subgroup P of G, let d be the smallest generator number of P and Ma（P） denote a family of maximal subgroups P1, P2 , Pd of P satisfying ∩^di=1 Pi...Let G be a finite group. Fix a prime divisor p of IGI and a Sylow p-subgroup P of G, let d be the smallest generator number of P and Ma（P） denote a family of maximal subgroups P1, P2 , Pd of P satisfying ∩^di=1 Pi = Ф（P）, the Frattini subgroup of P. In this paper, we shall investigate the influence of s-conditional permutability of the members of some fixed .Md（P） on the structure of finite groups. Some new results are obtained and some known results are generalized.展开更多
The relative xity of a permutation group is the maximum proportion of the points xed by a non-trivial element of the group,and the relative xity of a graph is the relative xity of its automorphism group,viewed as a pe...The relative xity of a permutation group is the maximum proportion of the points xed by a non-trivial element of the group,and the relative xity of a graph is the relative xity of its automorphism group,viewed as a permutation group on the vertex-set of the graph.We prove in this paper that the relative xity of connected 2-arc-transitive graphs of a xed valence tends to 0 as the number of vertices grows to in nity.We prove the same result for the class of arc-transitive graphs of a xed prime valence,and more generally,for any class of arc-transitive locally-L graphs,where L is a xed quasiprimitive graph-restrictive permutation group.展开更多
M?bius regular maps are surface embeddings of graphs with doubled edges such that(i)the automorphism group of the embedding acts regularly on flags and(ii) each doubled edge is a center of a M?bius band on the surface...M?bius regular maps are surface embeddings of graphs with doubled edges such that(i)the automorphism group of the embedding acts regularly on flags and(ii) each doubled edge is a center of a M?bius band on the surface. In this paper, we classify M?bius regular maps of order pq for any two primes p and q, where p≠q.展开更多
文摘In this paper, we investigate the algebraic structure of certain 2-generator groups of permutations of the integers. The groups fall into two infinite classes: one class terminates with the quaternion group and the other class terminates with the Klein-four group. We show that all the groups are finitely presented and we determine minimal presentations in each case. Finally, we determine the order of each group.
文摘Representation theory is concerned with the ways of explaining or visualizing a group as a group of matrices. In this paper, we extend the permutation pattern of to a two-line notation. We consider the representations of this non-deranged permutation group(p ≥ 5 and p a prime). Also we reveal some interesting properties and results of the character of where .
文摘Let n be a positive integer. A permutation a of the symmetric group of permutations of is called a derangement if for each . Suppose that x and y are two arbitrary permutations of . We say that a permutation a is a double derangement with respect to x and y if and for each . In this paper, we give an explicit formula for , the number of double derangements with respect to x and y. Let and let and be two subsets of with and . Suppose that denotes the number of derangements x such that . As the main result, we show that if and z is a permutation such that for and for , then where .
基金supported by the National Natural Science Foundation of China(Grant Nos.61201253 and 61303039)the Fundamental Research Funds for the Central Universities of China(Grant No.2682014CX095)
文摘The application of χ state are investigated in remote state preparation （RSP）. By constructing useful measurement bases with the aid of Hurwitz matrix equation, we propose several RSP schemes of arbitrary two- and three-qubit states via the χ state as the entangled resource. It is shown that the original state can be successfully prepared with the probability 100% and 50% for real coefficients and complex coefficients, respectively. For the latter case, the special ensembles with unit success probability are discussed by the permutation group. It is worth mentioning that the novel measurement bases have no restrictions on the coefficients of the prepared state, which means that the proposed schemes are more applicable.
文摘Some classical results about linear representations of a finite group G have been also proved for representations of G on non-abelian groups （G-groups）. In this paper we establish a decomposition theorem for irreducible G-groups which expresses a suitable irreducible G-group as a tensor product of two projective G-groups in a similar way to the celebrated theorem of Clifford for linear representations. Moreover, we study the non-abelian minimal normal subgroups of G in which this decomposition is possible.
基金Supported by the National Natural Science Foundation of China (Grant No.11071229)the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No.10KJD110004)the Postgraduate Innovation Grant of Xuzhou Normal University
文摘Let G be a finite group. Fix a prime divisor p of IGI and a Sylow p-subgroup P of G, let d be the smallest generator number of P and Ma（P） denote a family of maximal subgroups P1, P2 , Pd of P satisfying ∩^di=1 Pi = Ф（P）, the Frattini subgroup of P. In this paper, we shall investigate the influence of s-conditional permutability of the members of some fixed .Md（P） on the structure of finite groups. Some new results are obtained and some known results are generalized.
基金supported by the Austrian Science Fund(FWF)Project W1230-N13.The second author was supported by the Research Programme P1-0294the Research Project J1-1691,both funded by the Slovenian Research Agency(ARRS).
文摘The relative xity of a permutation group is the maximum proportion of the points xed by a non-trivial element of the group,and the relative xity of a graph is the relative xity of its automorphism group,viewed as a permutation group on the vertex-set of the graph.We prove in this paper that the relative xity of connected 2-arc-transitive graphs of a xed valence tends to 0 as the number of vertices grows to in nity.We prove the same result for the class of arc-transitive graphs of a xed prime valence,and more generally,for any class of arc-transitive locally-L graphs,where L is a xed quasiprimitive graph-restrictive permutation group.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11671276 and 11371259)
文摘M?bius regular maps are surface embeddings of graphs with doubled edges such that(i)the automorphism group of the embedding acts regularly on flags and(ii) each doubled edge is a center of a M?bius band on the surface. In this paper, we classify M?bius regular maps of order pq for any two primes p and q, where p≠q.
