A graph G is one-regular if its automorphism group Aut（G） acts transitively and semiregularly on the arc set. A Cayley graph Cay（Г, S） is normal if Г is a normal subgroup of the full automorphism group of Cay（...A graph G is one-regular if its automorphism group Aut（G） acts transitively and semiregularly on the arc set. A Cayley graph Cay（Г, S） is normal if Г is a normal subgroup of the full automorphism group of Cay（Г, S）. Xu, M. Y., Xu, J. （Southeast Asian Bulletin of Math., 25, 355-363 （2001）） classified one-regular Cayley graphs of valency at most 4 on finite abelian groups. Marusic, D., Pisanski, T. （Croat. Chemica Acta, 73, 969-981 （2000）） classified cubic one-regular Cayley graphs on a dihedral group, and all of such graphs turn out to be normal. In this paper, we classify the 4-valent one-regular normal Cayley graphs G on a dihedral group whose vertex stabilizers in Aut（G） are cyclic. A classification of the same kind of graphs of valency 6 is also discussed.展开更多
Denote by Dm the dihedral group of order 2m. Let R（Dm） be its complex representation ring, and let △（Dm） be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quoti...Denote by Dm the dihedral group of order 2m. Let R（Dm） be its complex representation ring, and let △（Dm） be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient △^n（Dm）/△^n＋1（Dm） for each positive integer n.展开更多
Let p be an odd prime, and D2p = （a,b I aP = b2 = l,bab= a 1） the dihedral group of order 2p. In this paper, we completely classify the cubic Cayley graphs on D2p up to isomorphism by means of spectral method. By th...Let p be an odd prime, and D2p = （a,b I aP = b2 = l,bab= a 1） the dihedral group of order 2p. In this paper, we completely classify the cubic Cayley graphs on D2p up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on D2p are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on D2 by using Gauss＇ celebrated law of quadratic reciprocity.展开更多
Cayley graphs have many good properties as models of communication networks. This study analyzes the reliability of the Cayley graph based on the dihedral graph. Graph theory and analyses show that almost all Cayley g...Cayley graphs have many good properties as models of communication networks. This study analyzes the reliability of the Cayley graph based on the dihedral graph. Graph theory and analyses show that almost all Cayley graphs of the dihedral graph D2n are optimal super-λ. The number Ni（G） of cutsets of size i, λ≤ i≤λ＇ is given as Ni（G） = n[^（n-1）δ i-δ].展开更多
We show that,up to isomorphism,there is a unique non-CI connected cubic Cayley graph on the dihedral group of order 2n for each even number n≥4.This answers in the negative the question of Li whether all connected cu...We show that,up to isomorphism,there is a unique non-CI connected cubic Cayley graph on the dihedral group of order 2n for each even number n≥4.This answers in the negative the question of Li whether all connected cubic Cayley graphs are CI-graphs(Discrete Math.,256,301-334(2002)).As an application,a formula is derived for the number of isomorphism classes of connected cubic Cayley graphs on dihedral groups,which generalises the earlier formula of Huang et al.dealing with the particular case when n is a prime(Acta Math.Sin.,Engl.Ser.,33,996-1011(2017)).As another application,a short proof is also given for a result on sparse circulant matrices obtained by Wiedemann and Zieve(arXiv preprint,(2007)).展开更多
Let D be a generalized dihedral group and Autcol（D） its Coleman automorphism group. Denote by Outcol（D） the quotient group of Autcol（D） by Inn（D）, where Inn（D） is the inner automorphism group of D. It is pro...Let D be a generalized dihedral group and Autcol（D） its Coleman automorphism group. Denote by Outcol（D） the quotient group of Autcol（D） by Inn（D）, where Inn（D） is the inner automorphism group of D. It is proved that either Outcol（D） = i or Outcol（D） is an elementary abelian 2-group whose order is completely determined by the cardinality of π（D）. Furthermore, a necessary and sufficient condition for Outcol（D） = 1 is obtained. In addition, whenever Outcol（D） ≠ 1, it is proved that Autcol（D） is a split extension of Inn（D） by an elementary abelian 2-group for which an explicit description is given.展开更多
Let G be a finite group and d a positive integer.Let s_(dN)(G)denote the smallest positive integer l such that every sequence over G of length at least l contains a nonempty product-one subsequence T with|T|≡0(mod d)...Let G be a finite group and d a positive integer.Let s_(dN)(G)denote the smallest positive integer l such that every sequence over G of length at least l contains a nonempty product-one subsequence T with|T|≡0(mod d).This paper studies s_(dN)(D_(2n))for the dihedral group D_(2n) and shows that when n=2^(r) with r≥3,the equality s_(dN)(D_(2n))=lcm(n,d)+gcd(n,d)holds.展开更多
In this paper, we completely determine the structure of the unit group of the group algebra of some dihedral groups D2 n over the finite field Fpk, where p is a prime.
The classification of groups of order less than 16 is reconsidered. The goal of the paper is partly historical and partly pedagogical and aims to achieve the classification as simply as possible in a way which can be ...The classification of groups of order less than 16 is reconsidered. The goal of the paper is partly historical and partly pedagogical and aims to achieve the classification as simply as possible in a way which can be easily incorporated into a first course in abstract algebra and without appealing to the Sylow Theorems. The paper concludes with some exercises for students.展开更多
Let V be a 2-dimensional vector space over the real field R with an affine or indefinite symmetric bilinear form. The infinite dihedral group W can be viewed as a subgroup of GL(V). In the present paper we will class...Let V be a 2-dimensional vector space over the real field R with an affine or indefinite symmetric bilinear form. The infinite dihedral group W can be viewed as a subgroup of GL(V). In the present paper we will classify all crystallographic groups associated with W up to conjugation in the affine group A(V).展开更多
In this paper, a certain class of welded knots K;is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K;,n ∈ Z;, we show that these welded knots are not equivalent to each oth...In this paper, a certain class of welded knots K;is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K;,n ∈ Z;, we show that these welded knots are not equivalent to each other and they are all not classical knots. Secondly, we study some properties of Gn and obtain that Gn is linear, residually finite and Hopfian.展开更多
Air Force Space Command is interested in improving the accuracy of GPS receiver positioning, navigation, and timing. To this end, it is useful to identify a set of optimal satellite constellations where each correspon...Air Force Space Command is interested in improving the accuracy of GPS receiver positioning, navigation, and timing. To this end, it is useful to identify a set of optimal satellite constellations where each corresponds to a configuration specifying the number of satellites in each orbital plane. These constellations could then be maintained in a library for future use as satellites fail and are launched. We utilize symmetry in the geometry of the GPS satellite orbits to partition the configurations into a much smaller set of equivalence classes where each class has the same overall receiver accuracy performance. We apply a classical algebraic combinatorial result, Polya＇s Theorem, to count and categorize the classes. Incorporating our results into a GPS constellation optimization computer tool will reduce run time by about an order of magnitude. We apply other algebraic and combinatorial techniques in original ways to count the class sizes and the classes that contain a given number of satellites. Finally, we break the equivalence classes into a still smaller set of new ＂structure＂ classes that are useful in applying the GPS computer tool.展开更多
Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. In this paper, we show that this conjecture is true for Cayley graph on generalized dihedral groups and generalized quaternion groups, ...Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. In this paper, we show that this conjecture is true for Cayley graph on generalized dihedral groups and generalized quaternion groups, which generalizes the result of F. Yang and X. Li [Inform. Process. Lett., 2011, 111： 416-419]. We also generalizes an early result of M. Nanasiova and M. Skoviera [J. Algebraic Combin., 2009, 30： 103-110].展开更多
Let k be an algebraically closed field of characteristic zero, and Dn be the dihedral group of order 2n, where n is a positive even integer. In this paper, we investigate Yetter-Drinfeld modules over the Hopf-Ore exte...Let k be an algebraically closed field of characteristic zero, and Dn be the dihedral group of order 2n, where n is a positive even integer. In this paper, we investigate Yetter-Drinfeld modules over the Hopf-Ore extension A（n,0） of kDn. We describe the structures and properties of simple Yetter Drinfeld modules over A（n, 0）, and classify all simple Yetter-Drinfeld modules over A（n, 0）.展开更多
We first study the spectrum of Hermitian adjacency matrix(H-spectrum)of Cayley digraphs X(D 2n,S)on dihedral group D2n with|S|=3.Then we show that all Cayley digraphs X(D2P,S)with|S|=3 and p odd prime are Cay-DS,namel...We first study the spectrum of Hermitian adjacency matrix(H-spectrum)of Cayley digraphs X(D 2n,S)on dihedral group D2n with|S|=3.Then we show that all Cayley digraphs X(D2P,S)with|S|=3 and p odd prime are Cay-DS,namely,for any Cayley digraph X(D2P,T),X(D2P,T)and X(D2P,S)having the same H-spectrum implies that they are isomorphic.展开更多
Let Fq be a finite field with order q and D2n be the dihedral group with 2n elements, and gcd(q, 2n) = 1. In this article, the authors give precise descriptions and enumerations of linear complementary dual(LCD) codes...Let Fq be a finite field with order q and D2n be the dihedral group with 2n elements, and gcd(q, 2n) = 1. In this article, the authors give precise descriptions and enumerations of linear complementary dual(LCD) codes and self-orthogonal codes in the finite dihedral group algebras Fq[D2n]. Some numerical examples are also presented to illustrate the main results.展开更多
In this paper,we completely classify a finite number of inequivalent indecom-posable modular representations of the dihedral group D2m in characteristic p,pl2m,and determine that there exists a summand of the group al...In this paper,we completely classify a finite number of inequivalent indecom-posable modular representations of the dihedral group D2m in characteristic p,pl2m,and determine that there exists a summand of the group algebra which is not a summand of the rings of coinvariants of these representations.展开更多
Let G be a finite group and S G\(e),where e denotes the identity element of G.The Cayley digraph X(G,S)of G over the set S is the graph with vertex set G and edge set((a,sa)I E G,s E S).Two graphs are called cospectra...Let G be a finite group and S G\(e),where e denotes the identity element of G.The Cayley digraph X(G,S)of G over the set S is the graph with vertex set G and edge set((a,sa)I E G,s E S).Two graphs are called cospectral if their adjacency matrices have the same spectrum.We construct a large family of cospectral non-isomorphic Cayley digraphs over the dihedral group of order 6p for p≥11.展开更多
In this paper, the Hopf Ore extension and corresponding module extension of the group algebra over dihedral group are studied. It turns out that the 1-dimensional and 2- dimensional simple representations can both be ...In this paper, the Hopf Ore extension and corresponding module extension of the group algebra over dihedral group are studied. It turns out that the 1-dimensional and 2- dimensional simple representations can both be extended to the simple representations over a class of Hopf Ore extension.展开更多
The paper investigates quantum stabilizer codes and our innovative construction methodology.At first,we began with representing fundamental concepts and theories of quantum computing.In order to identify quantum error...The paper investigates quantum stabilizer codes and our innovative construction methodology.At first,we began with representing fundamental concepts and theories of quantum computing.In order to identify quantum error correcting codes,we extend the method of using polynomials to represent qudits on square lattices to accommodate more complicated situations in general Cayley graphs.The paper briefly reviews essential definitions and examples related to graphs,groups,and stabilizer codes,and later we propose the novel method and demonstrate its application by using the dihedral group D_(4).展开更多
文摘A graph G is one-regular if its automorphism group Aut（G） acts transitively and semiregularly on the arc set. A Cayley graph Cay（Г, S） is normal if Г is a normal subgroup of the full automorphism group of Cay（Г, S）. Xu, M. Y., Xu, J. （Southeast Asian Bulletin of Math., 25, 355-363 （2001）） classified one-regular Cayley graphs of valency at most 4 on finite abelian groups. Marusic, D., Pisanski, T. （Croat. Chemica Acta, 73, 969-981 （2000）） classified cubic one-regular Cayley graphs on a dihedral group, and all of such graphs turn out to be normal. In this paper, we classify the 4-valent one-regular normal Cayley graphs G on a dihedral group whose vertex stabilizers in Aut（G） are cyclic. A classification of the same kind of graphs of valency 6 is also discussed.
文摘Denote by Dm the dihedral group of order 2m. Let R（Dm） be its complex representation ring, and let △（Dm） be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient △^n（Dm）/△^n＋1（Dm） for each positive integer n.
基金Supported by National Natural Science Foundation of China(Grant Nos.11671344 and 11531011)
文摘Let p be an odd prime, and D2p = （a,b I aP = b2 = l,bab= a 1） the dihedral group of order 2p. In this paper, we completely classify the cubic Cayley graphs on D2p up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on D2p are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on D2 by using Gauss＇ celebrated law of quadratic reciprocity.
基金Supported by the National Key Basic Research and Development (973) Program of China(No. 2007CB311003)
文摘Cayley graphs have many good properties as models of communication networks. This study analyzes the reliability of the Cayley graph based on the dihedral graph. Graph theory and analyses show that almost all Cayley graphs of the dihedral graph D2n are optimal super-λ. The number Ni（G） of cutsets of size i, λ≤ i≤λ＇ is given as Ni（G） = n[^（n-1）δ i-δ].
基金Supported by the Slovenian Research Agency (research program P1-0285 and research projects N1-0062,J1-9108,J1-1695,N1-0140,J1-2451,N1-0208 and J1-3001)。
文摘We show that,up to isomorphism,there is a unique non-CI connected cubic Cayley graph on the dihedral group of order 2n for each even number n≥4.This answers in the negative the question of Li whether all connected cubic Cayley graphs are CI-graphs(Discrete Math.,256,301-334(2002)).As an application,a formula is derived for the number of isomorphism classes of connected cubic Cayley graphs on dihedral groups,which generalises the earlier formula of Huang et al.dealing with the particular case when n is a prime(Acta Math.Sin.,Engl.Ser.,33,996-1011(2017)).As another application,a short proof is also given for a result on sparse circulant matrices obtained by Wiedemann and Zieve(arXiv preprint,(2007)).
基金Supported by a Discovery Grant from the Natural Science and Engineering Research Council of Canadathe National Natural Science Foundation of China(Grant Nos.71171120,71571108,11401329)+5 种基金the Project of International(Regional) Cooperation and Exchanges of NSFC(Grant No.71411130215)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20133706110002)the Natural Science Foundation of Shandong Province(Grant No.ZR2015GZ007)the Doctoral Fund of Shandong Province(Grant No.BS2012SF003)the Project of Shandong Province Higher Educational Science and Technology Program(Grant No.J14LI10)the Project of Shandong Province Higher Educational Excellent Backbone Teachers for International Cooperation and Training
文摘Let D be a generalized dihedral group and Autcol（D） its Coleman automorphism group. Denote by Outcol（D） the quotient group of Autcol（D） by Inn（D）, where Inn（D） is the inner automorphism group of D. It is proved that either Outcol（D） = i or Outcol（D） is an elementary abelian 2-group whose order is completely determined by the cardinality of π（D）. Furthermore, a necessary and sufficient condition for Outcol（D） = 1 is obtained. In addition, whenever Outcol（D） ≠ 1, it is proved that Autcol（D） is a split extension of Inn（D） by an elementary abelian 2-group for which an explicit description is given.
基金supported by the National Natural Science Foundation of China(No.12301425)。
文摘Let G be a finite group and d a positive integer.Let s_(dN)(G)denote the smallest positive integer l such that every sequence over G of length at least l contains a nonempty product-one subsequence T with|T|≡0(mod d).This paper studies s_(dN)(D_(2n))for the dihedral group D_(2n) and shows that when n=2^(r) with r≥3,the equality s_(dN)(D_(2n))=lcm(n,d)+gcd(n,d)holds.
文摘In this paper, we completely determine the structure of the unit group of the group algebra of some dihedral groups D2 n over the finite field Fpk, where p is a prime.
文摘The classification of groups of order less than 16 is reconsidered. The goal of the paper is partly historical and partly pedagogical and aims to achieve the classification as simply as possible in a way which can be easily incorporated into a first course in abstract algebra and without appealing to the Sylow Theorems. The paper concludes with some exercises for students.
文摘Let V be a 2-dimensional vector space over the real field R with an affine or indefinite symmetric bilinear form. The infinite dihedral group W can be viewed as a subgroup of GL(V). In the present paper we will classify all crystallographic groups associated with W up to conjugation in the affine group A(V).
基金The NSF(11329101,11431009,11301135 and 11471245) of China
文摘In this paper, a certain class of welded knots K;is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K;,n ∈ Z;, we show that these welded knots are not equivalent to each other and they are all not classical knots. Secondly, we study some properties of Gn and obtain that Gn is linear, residually finite and Hopfian.
文摘Air Force Space Command is interested in improving the accuracy of GPS receiver positioning, navigation, and timing. To this end, it is useful to identify a set of optimal satellite constellations where each corresponds to a configuration specifying the number of satellites in each orbital plane. These constellations could then be maintained in a library for future use as satellites fail and are launched. We utilize symmetry in the geometry of the GPS satellite orbits to partition the configurations into a much smaller set of equivalence classes where each class has the same overall receiver accuracy performance. We apply a classical algebraic combinatorial result, Polya＇s Theorem, to count and categorize the classes. Incorporating our results into a GPS constellation optimization computer tool will reduce run time by about an order of magnitude. We apply other algebraic and combinatorial techniques in original ways to count the class sizes and the classes that contain a given number of satellites. Finally, we break the equivalence classes into a still smaller set of new ＂structure＂ classes that are useful in applying the GPS computer tool.
基金Acknowledgements The first author was supported by the Natural Science Foundation of China （Grant No. 11301254）, the Natural Science Foundation of Henan Province （Grant No. 132300410313）, and the Natural Science Foundation of Education Bureau of Henan Province （Grant No. 13A110800）. The second author was supported by the National Natural Science Foundation of China （Grant No. 11171129） and the Doctoral Fund of Ministry of Education of China （Grant No. 20130144110001）.
文摘Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. In this paper, we show that this conjecture is true for Cayley graph on generalized dihedral groups and generalized quaternion groups, which generalizes the result of F. Yang and X. Li [Inform. Process. Lett., 2011, 111： 416-419]. We also generalizes an early result of M. Nanasiova and M. Skoviera [J. Algebraic Combin., 2009, 30： 103-110].
基金Supported by NSF of China (Grant No. 11171291)Doctorate Foundation (Grant No. 200811170001),Ministry of Education of China
文摘Let k be an algebraically closed field of characteristic zero, and Dn be the dihedral group of order 2n, where n is a positive even integer. In this paper, we investigate Yetter-Drinfeld modules over the Hopf-Ore extension A（n,0） of kDn. We describe the structures and properties of simple Yetter Drinfeld modules over A（n, 0）, and classify all simple Yetter-Drinfeld modules over A（n, 0）.
基金This work was partially supported by the National Natural Science Foundation of China(Nos.11561032 and 11771016)the Jiangxi Science Fund for Distinguished Young Scholars(No.20171BCB23032)also the Natural Science Foundation of Jiangxi Province(No.20192BAB201001).
文摘We first study the spectrum of Hermitian adjacency matrix(H-spectrum)of Cayley digraphs X(D 2n,S)on dihedral group D2n with|S|=3.Then we show that all Cayley digraphs X(D2P,S)with|S|=3 and p odd prime are Cay-DS,namely,for any Cayley digraph X(D2P,T),X(D2P,T)and X(D2P,S)having the same H-spectrum implies that they are isomorphic.
基金supported by the National Natural Science Foundation of China(Nos.61772015,11971321,12101326)Foundation of Nanjing Institute of Technology(No.CKJB202007)+4 种基金the NUPTSF(No.NY220137)the Guangxi Natural Science Foundation(No.2020GXNSFAA159053)the National Key Research and Development Program of China(No.2018YFA0704703)Foundation of Science and Technology on Information Assurance Laboratory(No.KJ-17-010)the Open Project of Shanghai Key Laboratory of Trustworthy Computing(No.OP202101)。
文摘Let Fq be a finite field with order q and D2n be the dihedral group with 2n elements, and gcd(q, 2n) = 1. In this article, the authors give precise descriptions and enumerations of linear complementary dual(LCD) codes and self-orthogonal codes in the finite dihedral group algebras Fq[D2n]. Some numerical examples are also presented to illustrate the main results.
基金NNSF(12171194)of ChinaNatural Science Foundation for Young Scientists of Shanxi Province(201901D211184)Natural Science Foundation for Young Scientists(12101375).
文摘In this paper,we completely classify a finite number of inequivalent indecom-posable modular representations of the dihedral group D2m in characteristic p,pl2m,and determine that there exists a summand of the group algebra which is not a summand of the rings of coinvariants of these representations.
基金L.H.Feng and W.J.Liu were supported by NSFC(Nos.12271527,12471022)L.Tang was supported by NSFC(No.12201202)Hunan Provincial Natural Science Foundation(Nos.2020JJ5096,2020JJ4233).
文摘Let G be a finite group and S G\(e),where e denotes the identity element of G.The Cayley digraph X(G,S)of G over the set S is the graph with vertex set G and edge set((a,sa)I E G,s E S).Two graphs are called cospectral if their adjacency matrices have the same spectrum.We construct a large family of cospectral non-isomorphic Cayley digraphs over the dihedral group of order 6p for p≥11.
基金the National Natural Science Foundation of China (No.10771182)
文摘In this paper, the Hopf Ore extension and corresponding module extension of the group algebra over dihedral group are studied. It turns out that the 1-dimensional and 2- dimensional simple representations can both be extended to the simple representations over a class of Hopf Ore extension.
文摘The paper investigates quantum stabilizer codes and our innovative construction methodology.At first,we began with representing fundamental concepts and theories of quantum computing.In order to identify quantum error correcting codes,we extend the method of using polynomials to represent qudits on square lattices to accommodate more complicated situations in general Cayley graphs.The paper briefly reviews essential definitions and examples related to graphs,groups,and stabilizer codes,and later we propose the novel method and demonstrate its application by using the dihedral group D_(4).
