Klein 4-group Research Articles

Computing superspecial hyperelliptic curves of genus 4 with automorphism group properly containing the Klein 4-group

In algebraic geometry or number theory, enumerating or finding superspecial curves in positive characteristic p is important both in theory and in computation. In this paper, we propose feasible algorithms to enumerate or find superspecial hyperelliptic curves of genus 4 with automorphism group properly containing the Klein 4-group. By executing the algorithms on Magma, we succeeded in enumerating such superspecial curves for all primes p with 19≤p<500, and in finding a single one for all primes p with 19≤p<7000.

Annihilator Ideals of Indecomposable Modules of Finite-Dimensional Pointed Hopf Algebras of Rank One

Let [Formula: see text] be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. We describe all annihilator ideals of indecomposable [Formula: see text]-modules by generators. In particular, we give the classification of all ideals of the finite-dimensional pointed rank one Hopf algebra of nilpotent type over the Klein 4-group.

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}.

KLEIN-4 İNVARYANTLARI ÜZERİNE

Let K[X_4 ]=K[x_1,x_2,x_3,x_4 ] be the polynomial algebra with 4 algebraically independent commuting variables over a field K of characteristic zero. The symmetric group S_4 acts on K[X_4 ] naturally by the action of permutations exchanging the indices of variables with respect to the corresponding permutation. It is well known that the algebra K[X_4 ]^S_4 of all polynomials preserved under the action of S_4 is generated by 4 algebraically independent elements called the elementary symmetric polynomials. In this study, we consider the subalgebra G of S_4 generated by the transpositions (13) and (24) which is isomorphic to the Klein-4 group, and find a free generating set for the algebra K[X_4 ]^G of G-invariants.

The relative Heller operator and relative cohomology for the Klein 4-group

Let G be the Klein four-group and let be an arbitrary field of characteristic 2. A classification of indecomposable -modules is known. We calculate the relative cohomology groups for every indecomposable -module N, where χ is the set of proper subgroups in G. This extends work of Pamuk and Yalcin to cohomology with non-trivial coefficients. We also show that all cup products in strictly positive degree in are trivial.

The interaction between a Q-fuzzy normal subgroup and a Q-fuzzy characteristic subgroup

Subsequently after the insertion of fuzzy sets by L. A. Zadeh indefinite number of researchers interpreted on the generalisation of the notion of fuzzy sets. Fuzzy sets span large arenas of research in engineering, medical sciences, social sciences, graph theory, etc. In this paper we introduce the concept of a Q-Fuzzy normal subgroup and Q-fuzzy characteristic subgroup and with the help of a Klein-4 Group we study the interaction between them.

Application of Klein-4 Group on Domino Card

Klein-4 or Klein-V group is a group with four elements including identity. The binary operation on Klein-4 group will produce identitiy if operated to it self and produce another non-identity element if operated to another non-identity element. The focus of this paper is to explain the principle of Klein-4 group on Domino card and also find all complete possible elements. Domino card is a set of 28 cards which the surface of each card divided in two identic boxes contain combination of dots as pattern. The final part of this research is to find out all possible combination of cards which can be used as elements of a Klein-4 groups.

Constructing the Standard Model fermions

The Standard Model has three generations of fermions and antifermions, each with two states of isospin, and each of these has both a lepton and a quark in three possible colour states. In total there are 48 states. No known system exists for constructing these from first principles. Here, it is suggested that the number of degrees of freedom required is a consequence of the nilpotent complexified vector-quaternion Dirac algebra, which emerges from the representation of the fundamental parameters mass, time, charge and space as a Klein-4 group, and that these degrees of freedom lead to unique structural representations of each of the individual fermions.

Locally finite derivations and modular coinvariants

We consider a finite dimensional kG-module V of a p-group G over a field k of characteristic p. We describe a generating set for the corresponding Hilbert Ideal. In case G is cyclic this yields that the algebra k[V]_G of coinvari-ants is a free module over its subalgebra generated by kG-module generators of V^∗ . This subalgebra is a quotient of a polynomial ring by pure powers of its variables. The coinvariant ring was known to have this property only when G was cyclic of prime order. In addition, we show that if G is the Klein 4-group and V does not contain an indecomposable summand isomorphic to the regular module, then the Hilbert Ideal is a complete intersection, extending a result of the second author and R. J. Shank.

The Spectrum of G- $${ { OD}}_{\lambda }(3,4,v)$$ O D λ ( 3 , 4 , v )

An ordered design $${ OD}_{\lambda }(t,k,v)$$ is a k-X array A, where $$|X|=v$$ , satisfying that each row of A contains k distinct elements of X and every t columns of A contain each ordered t-subset of X exactly $$\lambda $$ times. The subgroup of conjugation under which a k-X array A is invariant is called the conjugate invariant subgroup of A. If an $${ OD}_{\lambda }(t,k,v)$$ has a conjugate invariant subgroup G, then it is denoted by G- $${ OD}_{\lambda }(t,k,v)$$ . In this paper, we give the existence of G- $${ OD}_{\lambda }(3,4,v)~(\lambda \ge 2)$$ for $$C_{3}~ (\langle (123)(4)\rangle ), S_{3}$$ (symmetric group of degree 3), $$C_{4} ~(\langle (1234)\rangle )$$ and $$K_{4}$$ (Klein 4-group). Together with the known results, we almost completely determine the spectrum of G- $${ OD}_{\lambda }(3,4,v)$$ for G a subgroup of $$S_{4}$$ .

On semifields of order q4 with center \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \(\mathbb{F}_q\) \end{document}, admitting a Klein 4-group of automorphisms

The aim of this paper is to investigate the semifields of order q4 over a finite field of order q, q an odd prime power, admitting a Klein 4-group of automorphisms.

Tetravalent arc-transitive locally-Klein graphs with long consistent cycles

The topic of this paper is connected tetravalent graphs admitting an arc-transitive group of automorphisms G, such that the vertex-stabiliser Gv is isomorphic to the Klein 4-group. Such a graph will be called locally-Klein. A cycle in a graph is said to be consistent if there exists an automorphism of the graph that preserves the cycle set-wise and acts upon it as a one-step rotation. The main result of the paper is a classification of those locally-Klein graphs that contain a consistent cycle of length more than half the order of the graph. As a side result, we define an interesting family of graphs embedded on the torus or on the Klein bottle, such that the automorphism group of the resulting map has two orbits on the edges, two orbits on the vertices and two orbits on the arcs of the graph.

Automorphism groups of Cayley graphs generated by connected transposition sets

Let S be a set of transpositions that generates the symmetric group Sn, where n≥3. The transposition graph T(S) is defined to be the graph with vertex set {1,…,n} and with vertices i and j being adjacent in T(S) whenever (i,j)∈S. We prove that if the girth of the transposition graph T(S) is at least 5, then the automorphism group of the Cayley graph Cay(Sn,S) is the semidirect product R(Sn)⋊Aut(Sn,S), where Aut(Sn,S) is the set of automorphisms of Sn that fixes S. This strengthens a result of Feng on transposition graphs that are trees. We also prove that if the transposition graph T(S) is a 4-cycle, then the set of automorphisms of the Cayley graph Cay(S4,S) that fixes a vertex and each of its neighbors is isomorphic to the Klein 4-group and hence is nontrivial. We thus identify the existence of 4-cycles in the transposition graph as being an important factor in causing a potentially larger automorphism group of the Cayley graph.

An Extension Theorem for Terraces

We generalise an extension theorem for terraces for abelian groups to apply to non-abelian groups with a central subgroup isomorphic to the Klein 4-group $V$. We also give terraces for three of the non-abelian groups of order a multiple of 8 that have a cyclic subgroup of index 2 that may be used in the extension theorem. These results imply the existence of terraces for many groups that were not previously known to be terraced, including 27 non-abelian groups of order 64 and all groups of the form $V^s \times D_{8k}$ for all $s$ and all $k &gt; 1$ where $D_{8k}$ is the dihedral group of order $8k$.

The fourfold way of the genetic code

We describe a compact representation of the genetic code that factorizes the table in quartets. It represents a “least grammar” for the genetic language. It is justified by the Klein-4 group structure of RNA bases and codon doublets. The matrix of the outer product between the column-vector of bases and the corresponding row-vector V T = (C G U A), considered as signal vectors, has a block structure consisting of the four cosets of the K × K group of base transformations acting on doublet AA. This matrix, translated into weak/strong (W/S) and purine/pyrimidine (R/Y) nucleotide classes, leads to a code table with mixed and unmixed families in separate regions. A basic difference between them is the non-commuting (R/Y) doublets: AC/CA, GU/UG. We describe the degeneracy in the canonical code and the systematic changes in deviant codes in terms of the divisors of 24, employing modulo multiplication groups. We illustrate binary sub-codes characterizing mutations in the quartets. We introduce a decision-tree to predict the mode of tRNA recognition corresponding to each codon, and compare our result with related findings by Jestin and Soulé [Jestin, J.-L., Soulé, C., 2007. Symmetries by base substitutions in the genetic code predict 2′ or 3′ aminoacylation of tRNAs. J. Theor. Biol. 247, 391–394], and the rearrangements of the table by Delarue [Delarue, M., 2007. An asymmetric underlying rule in the assignment of codons: possible clue to a quick early evolution of the genetic code via successive binary choices. RNA 13, 161–169] and Rodin and Rodin [Rodin, S.N., Rodin, A.S., 2008. On the origin of the genetic code: signatures of its primordial complementarity in tRNAs and aminoacyl-tRNA synthetases. Heredity 100, 341–355], respectively.

On duality inducing automorphisms and sources of simple modules in classical groups

Let p be a prime number, k an algebraically closed field of characteristic p, P a finite p-group and W an indecomposable kP-module. Given a finite group G and an indecomposable kG-module V , we say that ðP; W Þ is a vertex–source pair for ðG; V Þ if there is an inclusion P ,! G of groups, under which P is a vertex of V and W is a source of V . Endo-permutation modules occur frequently as sources of simple modules of finite groups. For instance, every simple module of a p-soluble group has endopermutation source; if V is a simple kG-module, where G is a finite group, lying in a nilpotent block of kG, then V has endo-permutation source, and any 2-block of a finite group whose defect groups are isomorphic to the Klein 4-group possesses a simple module with endo-permutation source. By results of Berger and Feit, and independently Puig, the proof of which invokes the classification of finite simple groups, if W is an endo-permutation kP-module which occurs as a source of a simple kG-module, for a p-soluble group G, then the class of W is a torsion element in the Dade group of P. Further, Mazza [8] has shown that any endo-permutation kP-module whose isomorphism class is a torsion element of the Dade group of P and which satisfies certain structural constraints identified by Puig does occur as source of some simple module of some p-nilpotent group. By contrast, the question of which endo-permutation modules occur as sources of simple modules in simple, quasi-simple or almost simple groups has not been extensively studied. The smallest interesting case is the case where P is elementary abelian of rank 2, since if P is a finite cyclic group, there are no non-torsion endo-permutation kPmodules. In this paper, we study two situations in which simple modules of groups related to the finite classical groups having vertex–source pairs ðP; W Þ ,w ithP elementary abelian of order p 2 and W endo-permutation, occur and we prove that in both cases, W must be a self-dual module and hence corresponds to an element of order at most 2 in the corresponding Dade group.

More Mathematics in the Bedroom: A Paradoxical Probability

mattress turning: the four possible orientations for a mattress on a bed frame and the group of operations that move a mattress from one orientation to another. This group, isomorphic to the Klein 4-group, consists of the identity element and three 180? rota tions through the center of the mattress. I call these the flip (about an axis from head to foot), the flop (about an axis from side to side), and the spin (about a vertical axis). Of these, the flop is the most awkward to perform, and is perhaps most easily carried out as the composition of a flip with a spin?in either order, since the Klein 4-group is commutative. One part of the report particularly grabbed my attention:

Permutations of separable preference orders

The notion of separability is important in economics, operations research, and political science, where it has recently been studied within the context of referendum elections. In a referendum election on n questions, a voter's preferences may be represented by a linear order on the 2 n possible election outcomes. The symmetric group of degree 2 n , S 2 n , acts in a natural way on the set of all such linear orders. A permutation σ ∈ S 2 n is said to preserve separability if for each separable order ≻ , σ ( ≻ ) is also separable. Here, we show that the set of separability-preserving permutations is a subgroup of S 2 n and, for 4 or more questions, is isomorphic to the Klein 4-group. Our results indicate that separable preferences are rare and highly sensitive to small changes. The techniques we use have applications to the problem of enumerating separable preference orders and to other broader combinatorial questions.

Hyperelliptic Curves with Extra Involutions

AbstractThe purpose of this paper is to study hyperelliptic curves with extra involutions. The locusLgof such genus-ghyperelliptic curves is ag-dimensional subvariety of the moduli space of hyperelliptic curvesHg. The authors present a birational parameterization ofLgvia dihedral invariants, and show how these invariants can be used to determine the field of moduli of points p ∈ Lg. They conjecture that for p ∈Hgwith |Aut(p)| &gt; 2, the field of moduli is a field of definition, and they prove this conjecture for any point p ∈Lgsuch that the Klein 4-group is embedded in the reduced automorphism group ofp. Further, forg= 3, they show that for every moduli point p ∈H3such that |Aut(p)| &gt; 4, the field of moduli is a field of definition. A rational model of the curve over its field of moduli is provided.

Constructions of Hadamard Difference Sets

Using a spread ofPG(3, p) and certain projective two-weight codes, we give a general construction of Hadamard difference sets in groupsH×(Zp)4, whereHis either the Klein 4-group or the cyclic group of order 4, andpis an odd prime. In the casep≡3 (mod 4), we use an ovoidal fibration ofPG(3, p) to construct Hadamard difference sets, this construction includes Xia's construction of Hadamard difference sets as a special case. In the casep≡1 (mod 4), we construct new reversible Hadamard difference sets by explicitly constructing the two-weight codes needed in our general construction method. Using a well-known composition theorem, we conclude that there exist Hadamard difference sets with parameters (4m2, 2m2−m, m2−m), wherem=2a3b52c1132c2172c3p21p22…p2twitha, b, c1, c2, c3positive integers and where eachpjis a prime congruent to 3 modulo 4, 1⩽j⩽t.