Cyclic Group Of Order Research Articles

Spectral Networks and Stability Conditions for Fukaya Categories with Coefficients

Given a holomorphic family of Bridgeland stability conditions over a surface, we define a notion of spectral network which is an object in a Fukaya category of the surface with coefficients in a triangulated DG-category. These spectral networks are analogs of special Lagrangian submanifolds, combining a graph with additional algebraic data, and conjecturally correspond to semistable objects of a suitable stability condition on the Fukaya category with coefficients. They are closely related to the spectral networks of Gaiotto–Moore–Neitzke. One novelty of our approach is that we establish a general uniqueness results for spectral network representatives. We also verify the conjecture in the case when the surface is disk with six marked points on the boundary and the coefficients category is the derived category of representations of an A2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_2$$\\end{document} quiver. This example is related, via homological mirror symmetry, to the stacky quotient of an elliptic curve by the cyclic group of order six.

On parallelisms of PG(3,4) with automorphisms of order 2

Let PG(n,q) be the n-dimensional projective space over the finite field Fq. A spread in PG(n,q) is a set of mutually skew lines which partition the point set. A parallelism is a partition of the set of lines by spreads. The classification of parallelisms in small finite projective spaces is of interest for problems from projective geometry, design theory, network coding, error-correcting codes, and cryptography. All parallelisms of PG(3,2) and PG(3,3) are known. Parallelisms of PG(3,4) which are invariant under automorphisms of odd prime orders have also been classified. The present paper contributes to the classification of parallelisms in PG(3,4) with automorphisms of even order. We focus on cyclic groups of order four and the group of order two generated by a Baer involution. We examine invariants of the parallelisms such as the full automorphism group, the type of spreads and questions of duality. The results given in this paper show that the number of parallelisms of PG(3,4) is at least 8675365. Some future directions of research are outlined.

On an infinite family of integral Cayley graphs of Pauli groups

The classical Pauli group can be obtained as the central product of the dihedral group of 8 elements with the cyclic group of order 4. Inspired by this characterization, we introduce the notion of central product of Cayley graphs, which allows to regard the Cayley graph of a central product of groups as a quotient of the Cartesian product of the Cayley graphs of the factor groups. We focus our attention on the Cayley graph Cay(Pn,SPn) of the generalized Pauli group Pn on n-qubits; in fact, Pn may be decomposed as the central product of finite 2-groups, and a suitable choice of the generating set SPn allows us to recognize the structure of central product of graphs in Cay(Pn,SPn).Using this approach, we are able to recursively construct the adjacency matrix of Cay(Pn,SPn) for each n≥1, and to explicitly describe its spectrum and the associated eigenvectors. It turns out that Cay(Pn,SPn) is a (3n+2)-regular bipartite graph on 4n+1 vertices, and it has integral spectrum. This is a highly nontrivial property if one considers that, by choosing as a generating set for P1 the three classical Pauli matrices, one gets the so-called Möbius-Kantor graph, belonging to the class of generalized Petersen graphs, whose spectrum is not integral.

Order Structured Graphs of Cyclic Groups and their Classification

Let $\Gamma^{o}(G)$ with $G\cong C_{p},$ a cyclic group of order $p,$ be an order structured graph. The group $C_{p}$ will be assumed as the vertex set of the graph $\Gamma^{o}(G)$ and an edge between vertices will be built on the basis of a defined relation via order structure. Certain graphical parameters such as independence ratio, clique number, domination number, and separability are discussed. Some characterizations are proposed and proved by incorporating the defined relation. It is further proved that $\Gamma^{o}(C_{p})$ can never be a hamiltonian graph. Lastly, It is shown that $C(\Gamma^{o}(C_{p}))$ is isomorphic to $\Gamma^{o}(C_{p}).

On the Unit Group of the Integral Group Ring Z(S_3×C_3)

Describing the group of units in the integral group ring is a famous and classical open problem. Let S_3 and C_3 be the symmetric group of order 6 and a cyclic group of order 3, respectively. In this paper, a description of the units of the integral group ring Z(S_3×C_3) of the direct product group S_3×C_3 concerning a complex representation of degree two is given. As a result, a part of the conjecture which is introduced in (Low, 2008) and related to group rings over a complex integral domain is resolved using representation theory.

A note on one decomposition of the 2-primary part of K 2(ℤ[C p × (C 2) n

Let Cn be a cyclic group of order n. We investigate K 2 of integral group rings via the Mayer-Vietoris sequence, and give a decomposition of the 2-primary torsion subgroup of K 2 ( Z [ C p × ( C 2 ) n ] ) for any prime p ≡ 3 , 5 , 7 ( mod 8 ) , in particular, K 2 ( Z [ C 3 × ( C 2 ) n ] ) is proven to be a finite abelian 2-group. As an application, we prove K 2 ( Z [ C 3 × ( C 2 ) 2 ] ) is an elementary abelian 2-group of rank at least 14, at most 16.

On the Terwilliger algebra of the group association scheme of Cn⋊C2

In 1992, Terwilliger introduced the notion of the Terwilliger algebra in order to study association schemes. The Terwilliger algebra of an association scheme A is the subalgebra of the complex matrix algebra, generated by the Bose-Mesner algebra of A and its dual idempotents with respect to a point x.In Bannai and Munemasa (1995) [3] determined the dimension of the Terwilliger algebra of abelian groups and dihedral groups, by showing that they are triply transitive (i.e., triply regular and dually triply regular). In this paper, we give a generalization of their results to the group association scheme of semidirect products of the form Cn⋊C2, where Cm is a cyclic group of order m≥2. Moreover, we will give the complete characterization of the Wedderburn components of the Terwilliger algebra of these groups.

Characterization of the unit group of ℤ[G × C2n

Let [Formula: see text] be a group and [Formula: see text] be the cyclic group of order 2. We characterize the unit group [Formula: see text] of the integral group ring [Formula: see text] in terms of the unit group [Formula: see text], for every positive integer [Formula: see text]. As a consequence, for each [Formula: see text], we determine conditions on some class of matrices [Formula: see text]s, to be in GL[Formula: see text]. Using this we calculate the integer solutions of some specific class of symmetric polynomial [Formula: see text] with integer coefficients.

Efficient Search for Superspecial Hyperelliptic Curves of Genus Four with Automorphism Group Containing C6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{C}}_6$$\\end{document}

In arithmetic and algebraic geometry, superspecial (s.sp. for short) curves are one of the most important objects to be studied, with applications to cryptography and coding theory. If g≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g \\ge 4$$\\end{document}, it is not even known whether there exists such a curve of genus g in general characteristic p>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p > 0$$\\end{document}, and in the case of g=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g=4$$\\end{document}, several computational approaches to search for those curves have been proposed. In the genus-4 hyperelliptic case, Kudo-Harashita proposed a generic algorithm to enumerate all s.sp. curves, and recently Ohashi-Kudo-Harashita presented an algorithm specific to the case where automorphism group contains the Klein 4-group as a subgroup. In this paper, we propose an algorithm with complexity O~(p4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ilde{O}}(p^4)$$\\end{document} in theory but O~(p3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ilde{O}}(p^3)$$\\end{document} in practice to enumerate s.sp. hyperelliptic curves of genus 4 with automorphism group containing the cyclic group of order 6. By executing the algorithm over Magma, we enumerate those curves for p up to 1000. We also succeeded in finding a s.sp. hyperelliptic curve of genus 4 in every p with p≡2(mod3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\equiv 2 \\pmod {3}$$\\end{document}.

Left regular bands of groups and the Mantaci–Reutenauer algebra

We develop the idempotent theory for algebras over a class of semigroups called left regular bands of groups (LRBGs), which simultaneously generalize group algebras of finite groups and left regular band (LRB) algebras. Our techniques weave together the representation theory of finite groups and LRBs, opening the door for a systematic study of LRBGs in an analogous way to LRBs. We apply our results to construct complete systems of primitive orthogonal idempotents in the Mantaci–Reutenauer algebra MRn[G] associated to any finite group G. When G is abelian, we give closed form expressions for these idempotents, and when G is the cyclic group of order two, we prove that they recover idempotents introduced by Vazirani.

On zero-sum free sequences contained in random subsets of finite cyclic groups

Let Cn be a cyclic group of order n. A sequence S of length ℓ over Cn is a sequence S=a1⋅a2⋅…⋅aℓ of ℓ elements in Cn, where a repetition of elements is allowed and their order is disregarded. We say that S is a zero-sum sequence if Σi=1ℓai=0 and that S is a zero-sum free sequence if S contains no zero-sum subsequence. In 2000, Gao obtained a construction of all zero-sum free sequences of length n−1−k over Cn for 0≤k≤n3. In this paper, we consider a generalization for a random subset of Cn. Let R=R(Cn,p) be a random subset of Cn obtained by choosing each element in Cn independently with probability p. Let Nn−1−kR be the number of zero-sum free sequences of length n−1−k in R. Also, let Nn−1−k,dR be the number of zero-sum free sequences of length n−1−k having d distinct elements in R. We obtain the expectations of Nn−1−kR and Nn−1−k,dR for 0≤k≤n3 and show that Nn−1−kR and Nn−1−k,dR are asymptotically almost surely (a.a.s.) concentrated around their expectations when k is fixed. Moreover, we provide two ways to compute the expectations using partition numbers and a recurrence formula.

Değişmeli Grup Halkalarında G-Nilpotent Birimsel Elemanların Direkt Çarpım Gruplarına Bir Genellemesi

Let V(RG) denote the normalized unit group of the group ring RG of a group G over a ring R. The concept of G-nilpotent unit in a commutative group ring has been defined in (Danchev, 2012). In this study, some necessary and sufficient conditions for a normalized unit group in a commutative group ring of a direct product group G×H to consist only of G×H-nilpotent units have been given and especially some results which are related to groups G×C_3 and G×C_4 have been introduced where C_3 and C_4 are cyclic groups of orders 3 and 4 respectively. In this context, we can say that the paper extends the results in (Danchev, 2012). At the end, an open problem is served as a future work.

On the weight of finite groups

For a finite group G, let W(G) denotes the set of the orders of the elements of G. In this paper we study jW(G)j and show that the cyclic group of order n has the maximum value of jW(G)j among all groups of the same order. Furthermore we study this notion in nilpotent and non-nilpotent groups and state some inequality for it. Among the result we show that the minimum value of jW(G)j is power of 2 or it pertains to a non-nilpotent group.

A note on Bass' conjecture

For a finite group G, we denote by d(G) and by E(G), respectively, the small Davenport constant and the Gao constant of G. Let Cn be the cyclic group of order n and let Gm,n,s=Cn⋊sCm be a metacyclic group. In [2, Conjecture 17], Bass conjectured that d(Gm,n,s)=m+n−2 and E(Gm,n,s)=mn+m+n−2 provided ordn(s)=m. In this paper, we show that the assumption ordn(s)=m is essential and cannot be removed. Moreover, if we suppose that Bass' conjecture holds for Gm,n,s and the mn-product-one free sequences of maximal length are well behaved, then Bass conjecture also holds for G2m,2n,r, where r2≡s(modn).

On the direct and inverse zero-sum problems over Cn⋊sC2

Let Cn be the cyclic group of order n. In this paper, we provide the exact values of some zero-sum constants over G=Cn⋊sC2 where s≢±1(modn), namely η-constant, Gao constant, and Erdős-Ginzburg-Ziv constant (the latter for all but a “small” family of cases). As a consequence, we prove the Gao's and Zhuang-Gao's Conjectures for groups of this form. We also solve the associated inverse problems by characterizing the structure of the sequences S1,S2,S3 over G of maximal lengths such that S1 has no short product-one subsequences, S2 has no product-one subsequences of length exp⁡(G), and S3 has no product-one subsequences of length |G|.

Influence of the Baer–Kaplansky Theorem on the Development of the Theory of Groups, Rings, and Modules

The review presents an analysis of results of the Abelian group theory, as well as rings and modules, which concern the definability of algebraic structures by their endomorphism rings and related structures. In the systematization of the results, the greatest attention is paid to torsion-free Abelian groups, which are of particular interest due to the presence of non-isomorphic direct decompositions in this class. This significantly expands the understanding of general, including modern, trends of the development of algebra in the context related to the Baer–Kaplansky theorem. The reflection of the properties of algebraic objects of a certain class in their endomorphism rings is a natural structural connection, the study of which is a separate investigation direction. A striking introduction to this topic was the Baer–Kaplansky theorem for torsion Abelian groups, which dates back to the middle of the last century and states that any isomorphism of endomorphism rings of two groups from this class is inevitably induced by some isomorphism of the groups themselves. Of course, it follows that if two torsion Abelian groups have isomorphic endomorphism rings, then they are isomorphic. This profound result inspired mathematicians to obtain results in the same form concerning other classes of objects. But even in the theory of Abelian groups itself, other classes were discovered for which the analogue of the Baer–Kaplansky theorem is valid. Despite the fundamental difference between the definitions of completely decomposable Abelian groups, which are direct sums of rank-one torsion-free groups, and torsion Abelian groups considered, which are direct sums of finite order cyclic groups, there is one very important common characteristic of these classes: their decompositions into indecomposable summands are determined uniquely up to isomorphism. This property is not possessed by torsion-free Abelian groups in general, whose definability by their endomorphism rings is in the focus of our attention.

Central automorphisms of Zappa-Szép products of two cyclic groups

The central automorphism group of the Zappa-Sz?p product of two cyclic groups of orders m and p2 is calculated, where p is a prime.

Automorphism groups of bipartite Kneser type-k graphs

For integers [Formula: see text], [Formula: see text], let [Formula: see text]. Let [Formula: see text] be the set of all non-empty subsets of [Formula: see text]. Let [Formula: see text] be the set of [Formula: see text]-element subsets of [Formula: see text], and [Formula: see text]. A bipartite Kneser type-[Formula: see text] graph [Formula: see text] is defined with parts [Formula: see text] and [Formula: see text], and a vertex [Formula: see text] is adjacent to a vertex [Formula: see text] if and only if [Formula: see text] or [Formula: see text]. The algebraic properties of bipartite Kneser type-[Formula: see text] graphs are investigated. For any integers [Formula: see text], the automorphism groups of bipartite Kneser type-1 graphs are isomorphic to the symmetric group [Formula: see text]. The bipartite Kneser type-1 graph’s Wiener index, peripheral Wiener index, and peripheral Hosoya polynomial were established. For integers [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] or [Formula: see text], [Formula: see text], the automorphism group of [Formula: see text] is isomorphic to the symmetric group [Formula: see text]. For [Formula: see text] is even and [Formula: see text], the automorphism group of [Formula: see text] is isomorphic to [Formula: see text], where [Formula: see text] is the cyclic group of order 2.

Wedderburn decomposition of a semisimple group algebra $\mathbb{F}_qG$ from a subalgebra of factor group of $G$

In this paper, we derive a condition under which the Wedderburn decomposition of a semisimple group algebra $\mathbb{F}_qG$ can be deduced from the Wedderburn decomposition of $\mathbb{F}_q(G/H)$, where $H$ is a normal subgroup of $G$ having two elements and $q=p^k$ for some prime $p$ and $k\in \mathbb{Z}^+$. In order to complement the abstract theory of the paper, we deduce the Wedderburn decomposition and hence the unit group of semisimple group algebra $\mathbb{F}_q(A_5\rtimes C_4)$, where $A_5\rtimes C_4$ is a non-metabelian group and $C_4$ is a cyclic group of order $4$.

The existence of 𝔽 q -primitive points on curves using freeness

Let 𝒞 Q be the cyclic group of order Q, n a divisor of Q and r a divisor of Q/n. We introduce the set of (r,n)-free elements of 𝒞 Q and derive a lower bound for the number of elements θ∈𝔽 q for which f(θ) is (r,n)-free and F(θ) is (R,N)-free, where f,F∈𝔽 q [x]. As an application, we consider the existence of 𝔽 q -primitive points on curves like y n =f(x) and find, in particular, all the odd prime powers q for which the elliptic curves y 2 =x 3 ±x contain an 𝔽 q -primitive point.