Dicyclic Groups Research Articles

The 1-rotational Kirkman triple systems of order 33

We establish that there are exactly 500 KTS(33)s admitting an automorphism group fixing one point and acting regularly on the remainder; 436 are over the cyclic group while 64 are over the dicyclic group. There are exactly 243 nonisomorphic STS(33)s underlying the above KTS(33)s; 211 are over the cyclic group while 32 are over the dicyclic group. This gives a significant improvement on the number of known KTS(33)s (at least 528 instead of at least 28).

On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group

A necessary and sufficient condition for the existence of orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with the dicyclic group is given. In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed.

Nonabelian discrete symmetries, fermion mass textures and large neutrino mixing

Nonabelian discrete groups are an attractive tool to describe fermion masses and mixings. They have nonsinglet representations which seem particularly suitable for distinguishing the lighter generations from the heavier ones. Also, they do not suffer from the extra constraints a continuous group must obey, e.g. limits on extra particles. Some of the simplest groups are the nonabelian discrete subgroups of SO(3) and SU(2), the so called dihedral groups D n and dicyclic groups Q 2 n , which both have only singlet and doublet representations. After studying which vacuum expectation value (VEV) directions of representations of dihedral and dicyclic groups preserve which subgroups, we construct a simple model based on the group Q 6× Q 6. The model reproduces the masses and mixings of all quarks and leptons, including neutrinos. It has a large mixing angle in the μ− τ neutrino sector, in accordance with the recent SuperKamiokande results, while keeping a small quark mixing in the bottom–charm sector. The reason is similar to the one found in the literature based on the SU(5) group: the large left handed mixing angle in the lepton sector corresponds to the large unphysical right handed in the down quark sector. The large mixing is also responsible for the different hierarchies of the two heaviest families in the up and down sector, and can be summarized as the order of magnitude relation: m s m b ∼ tan(θ μτ) m c m t .

Rotational invariance of two-level group codes over dihedral and dicyclic groups

Phase-rotational invariance properties for two-level constructed, (using a binary code and a code over a residue class integer ring as component codes) Euclidean space codes (signal sets) in two and four dimensions are discussed. The label codes are group codes over dihedral and dicyclic groups respectively. A set of necessary and sufficient conditions on the component codes is obtained for the resulting signal sets to be rotationally invariant to several phase angles.

Frame starters and adders in dicyclic groups

Frame starters and adders in dicyclic groups

The class number one problem for some non-abelian normal CM-fields

Let N be a non-abelian normal CM-eld of degree 4p; p any odd prime. Note that the Galois group of N is either the dicyclic group of order 4p; or the dihedral group of order 4p: We prove that the (relative) class number of a dicyclic CM-eld of degree 4p is always greater then one. Then, we determine all the dihedral CM-elds of degree 12 with class number one: there are exactly nine such CM-elds.

A note on frame starters

We construct frame starters in Z2n − {0, n}, for n ≡ 0, 1 mod 4, where Z2n denotes the cyclic group of order 2n. We also construct left frame starters in Q2n − {e, αn}, where Q2n is the dicyclic group of order 4n and αn is the unique element of order 2 in Q2n.

Dicyclic horizontal symmetry and supersymmetric grand unification.

It is shown how to use as horizontal symmetry the dicyclic group ${Q}_{6}\ensuremath{\subset}\mathrm{SU}(2)$ in a supersymmetric unification SU(5)\ensuremath{\bigotimes}SU(5)\ensuremath{\bigotimes}SU(2) where one SU(5) acts on the first and second families, in a horizontal doublet, and the other acts on the third. This can lead to acceptable quark masses and mixings, with an economic choice of matter supermultiplets, and charged lepton masses can be accommodated.

SIMPLE NON-ABELIAN FINITE FLAVOR GROUPS AND FERMION MASSES

The use of non-Abelian discrete groups G as family symmetries is discussed in detail. Out of all such groups up to order g=31, the most appealing candidates are two subgroups of SU(2): the dicyclic (double dihedral) group G=Q6= (d) D3(g=12) and the double tetrahedral group [Formula: see text]. Both can allow a hierarchy t>b, τ>c>s, μ>u, d, e. The top quark is uniquely allowed to have a G symmetric mass. Sequential breaking of G and radiative corrections give the smaller masses. Anomaly freedom for gauging G⊂ SU (2) is a strong constraint in assignment of fermions to representations of G.

Quark mass textures within a finite non-Abelian dicyclic group.

Using as a flavor symmetry a finite nonabelian dicyclic $Q_{2N}$ group, we show how to derive quark mass matrices with two arrangements of symmetric texture zeros which are phenomenologically viable. Three other such acceptable textures in the recent literature are unattainable in this approach and hence disfavored. We assume massive vector-like fermions and Higgs singlets transforming as judiciously-chosen $Q_{2N}$ doublets and use the tree-level mass generation mechanism of Froggatt and Nielsen.

Quark Mass Textures within a Finite Non-Abelian Dicyclic Group

Quark Mass Textures within a Finite Non-Abelian Dicyclic Group

A Lower Bound for the Real Genus of a Finite Group

Abstract Let G be a finite group. The real genus ρ(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we obtain a good general lower bound for the real genus of the group G. We use the standard representation of G as a quotient of a non-euclidean crystallographic group by a bordered surface group. The lower bound is used to determine the real genus of several infinite families of groups; the lower bound is attained for some of these families. Among the groups considered are the dicyclic groups and some abelian groups. We also obtain a formula for the real genus of the direct product of an elementary abelian 2-group and an “even” dicyclic group. In addition, we calculate the real genus of an abstract family of groups that includes some interesting 3-groups. Finally, we determine the real genus of the direct product of an elementary abelian 2-group and a dihedral group.

Finite metacyclic groups acting on bordered surfaces

A group is called metacyclic in case both its commutator subgroup and commutator quotient group are cyclic. Thus a metacyclic group is a cyclic extension of a cyclic group, and metacyclic groups are among the best understood of the nonabelian groups. Many interesting groups are metacyclic. For instance, the dihedral groups and the “odd” dicyclic groups are metacyclic; see [4, pp. 9–11] for more examples. Here we shall consider the actions of these groups on bordered Klein surfaces.

On the R-sequenceability of dicyclic groups

We give a sufficient condition for the R-sequenceability of the dicyclic group Q 2 n = gp{ a, b: a 2 n = e, b 2= a n , ab= ba -1} of order 4 n and give two specific constructions by which the condition can be met when (1) n=4 k or (2) n + 1 is a prime p say and -2 is a primitive root modulo p. We conjecture that the condition can be met for all even integers n>2.

On the harmoniousness of dicyclic groups

A group is called harmonious if its elements can be arranged in a sequence such that the products of consecutive elements comprise all elements of the group. We give a sufficient condition for the dicyclic group of order 4 n to be harmonious and give a specific construction by which the condition can be met when n is divisible by 4.

Neutrino magnetic moment and the dicyclic group.

We show a mechanism using a discrete symmetry, the dicyclic group ${\mathrm{Q}}_{6}$, which naturally gives rise to a small mass but a large transitional magnetic moment for the neutrino such that the anticorrelation between the solar neutrino intensity and the sunspot activity can be explained. The process of muonium-antimuonium oscillation provides the most interesting constraint to the model besides neutrino physics.

Quaternionic and octonionic orbifolds

We identify the four- and six-dimensional orbifolds embeddable in E 8, the root system of which is constructed with octonions and icosians. The root system of E 6 expressed as integral octonions can be interpreted as a collection of the binary tetrahedral groups and the dicyclic groups of order 12.

On 1-factorizability of Cayley graphs

In this paper we consider the existence of a 1-factorization of undirected Cayley graphs of groups of even order. We show that a 1-factorization exists for all generating sets for even order abelian groups, dihedral groups, and dicyclic groups and for all minimal generating sets for even order nilpotent groups and for D m × Z n . We also derive other results that are useful in considering specific Cayley graphs. These results support the conjecture that all Cayley graphs of groups of even order are 1-factorizable. If this is not the case the same result may hold for minimal generating sets.

On hamiltonian circuits in cartesian products of Cayley digraphs

In this paper, we investigate the existence of a hamiltonian circuit in the cartesian product of two Cayley digraphs. Three of our results can be summarized as follows. Suppose K is the Cayley digraph of a dihedral, semidihedral, or dicyclic group arising from a specified pair of (standard) generators, and suppose L is a Cayley digraph with a hamiltonian circuit. Then, the cartesian product of K and L has a hamiltonian circuit. As a corollary to our main theorem, we also show that the cartesian product of an undirected cycle of length n and a directed cycle of length k has a hamiltonian circuit unless n = 2 and k is odd. Some open problems are stated.

Topological Markov chains with dicyclic dimension group.

Topological Markov chains with dicyclic dimension group.