Quaternion Group Research Articles

Hurwitz Equivalence in Tuples of Generalized Quaternion Groups and Dihedral Groups

Let $Q_{2^m}$ be the generalized quaternion group of order $2^m$ and $D_N$ the dihedral group of order $2N$. We classify the orbits in $Q_{2^m}^n$ and $D_{p^m}^n$ ($p$ prime) under the Hurwitz action.

Hochschild Cohomology Ring of an Order of a Simple Component of the Rational Group Ring of the Generalized Quaternion Group

We will give an efficient bimodule projective resolution of an order Γ, where Γ is an order of a simple component of the rational group ring ℚ Q 2 r of the generalized quaternion 2-group Q 2 r of order 2 r+2. Moreover, we will determine the ring structure of the Hochschild cohomology HH*(Γ) by calculating the Yoneda products using this bimodule projective resolution.

On the number of subgroups of given type in a finite p-group

In §1 we study the p-groups G containing exactly p + 1 subgroups of order p p and exponent p. A number of counting theorems and results on subgroups of maximal class and p-groups with few subgroups of given type are also proved. Counting theorems play crucial role in the whole paper. This paper is a continuation of (Ber1, Ber3, Ber4, BJ2). We use the same notation, however, for the sake of convenience, we recall it in the following paragraph. In what follows, p is a prime, n, m, k, s, t are natural numbers, G is a finite p-group of order |G|, o(x) is the order of x ∈ G, n(G) = h x ∈ G | o(x) ≤ p n i , ∗(G) = h x ∈ G | o(x) = p n i and ℧n(G) = h x p n | x ∈ Gi . A p-group G is said to be absolutely regular if |G/℧1(G)| 2 and cl(G) = m − 1. As usually, G ' , �(G), Z(G) denote the derived subgroup, Frattini subgroup and center of G, respectively. Let i = {H 2 and n > 3 for p = 2, let Mpn = h a, b | a p n 1 = b p = 1, a b = a 1+p n 2 i . Let D2m, Q2m and SD2m be dihedral, generalized quaternion and semidihedral groups of order 2 m , and let Cpn, Epn be cyclic and elementary abelian groups

The classification and the conjugacy classes of the finite subgroups of the sphere braid groups

We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups $P_{n}(RP^2)$ of the projective plane. The maximal finite subgroups of $P_{n}(RP^2)$ are isomorphic to the quaternion group of order 8 if $n=3$, and to $\Z_{4}$ if $n\geq 4$. Further, for all $n\geq 3$, up to isomorphism, the following groups are the infinite virtually cyclic subgroups of $P_{n}(RP^2)$: $\Z$, $\Z_{2} \times \Z$ and the amalgamated product $\Z_{4} \ast_{\Z_{2}} \Z_{4}$.

A separable deformation of the quaternion group algebra

The Donald-Flanigan conjecture asserts that for any finite group G and any field k, the group algebra kG can be deformed to a separable algebra. The minimal unsolved instance, namely the quaternion group Q8 over a field k of characteristic 2 was considered as a counterexample. We present here a separable deformation of kQ8. In a sense, the conjecture for any finite group is open again.

Examples of exotic free 2–complexes and stably free nonfree modules for quaternion groups

This is a continuation of our study [3] of a family of projective modules over Q4n , the generalized quaternion (binary dihedral) group of order 4n. Our approach is constructive. Whenever n 7 is odd, this work provides examples of stably free nonfree modules of rank 1, which are then used to construct exotic algebraic 2‐ complexes relevant to Wall’s D(2)‐problem. While there are examples of stably free nonfree modules for many infinite groups G , there are few actual examples for finite groups. This paper offers an infinite collection of finite groups with stably free nonfree modules P , given as ideals in the group ring. We present a method for constructing explicit stabilizing isomorphisms W ZG ZGaP ZG described by 2 2 matrices. This makes the subject accessible to both theoretical and computational investigations, in particular, of Wall’s D(2)‐problem. 16D40, 19A13, 57M20; 55P15

An extraordinary origami curve

AbstractWe study the origami W defined by the quaternion group of order 8 and its Teichmüller curve C (W) in the moduli space M3. We prove that W has Veech group SL2(ℤ), determine the equation of the family over C (W) and find several further properties. As main result we obtain infinitely many origami curves in M3 that intersect C (W). We also present a combinatorial description of these origamis. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The spectrum of the Dirac operator on SU2/Q8

We compute the fundamental Dirac operator for the three-parameter-family of homogeneous Riemannian metrics and the four different spin structures on SU2/Q8, where Q8 denotes the group of quaternions. We deduce its spectrum for the Berger metrics and show the sharpness of Christian Bar’s upper bound for the smallest Dirac eigenvalue in the particular case where SU2/Q8 is a homogeneous minimal hypersurface of S4.

Conjugacy classes of non-normal subgroups in finite nilpotent groups

Let v(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. Poland and Rhemtulla [J. Poland and A. Rhemtulla. The number of conjugacy classes of non-normal subgroups in nilpotent groups. Comm. Algebra24 (1996), 3237–3245.] proved that if G is nilpotent of class c then c − 1 unless G is a Hamiltonian group. The sharpness of this lower bound is a problem about finite p-groups, since , where μ(G) denotes the number of normal subgroups of G. In this paper, we show that the bound c − 1 can be substantially improved: if G is a finite p-group and , then p(k − 1) + 1, unless G is a Hamiltonian group or a generalized quaternion group. In these exceptional cases, G is a 2-group and 2(k − 1).

Common origin ofθ13andΔm122in a model of neutrino mass with quaternion symmetry

The smallness of the 1--3 lepton mixing angle ${\ensuremath{\theta}}_{13}$ and of the neutrino mass-squared-difference ratio $\ensuremath{\Delta}{m}_{12}^{2}/\ensuremath{\Delta}{m}_{23}^{2}$ can be understood as the departure from a common limit where they both vanish. We discuss in general the conditions for realizing the mass degeneracy of a pair of neutrinos and show that the vanishing of a $CP$ violating phase is needed. We find that the discrete quaternion group $Q$ of eight elements is the simplest family symmetry which correlates the smallness of $\ensuremath{\Delta}{m}_{12}^{2}$ to the value of ${\ensuremath{\theta}}_{13}$. In such a model we predict $0.12\ensuremath{\lesssim}\mathrm{sin}{\ensuremath{\theta}}_{13}\ensuremath{\lesssim}0.2$ if the ordering of the neutrino mass spectrum is normal, and $\mathrm{sin}{\ensuremath{\theta}}_{13}\ensuremath{\lesssim}0.12$ if it is inverted.

Classes de Steinitz d'extensions quaternioniennes généralisées de degré 4p r

AbstractLet p be an odd prime number, r a natural number, and H4pr the generalized quaternion group of order 4p r . Let k be a number field and Cl(k) its class group. Let Rm(k,H4pr) be the subset of Cl(k) consisting of those classes which are realizable as Steinitz classes of tame Galois extensions of k with Galois group isomorphic to H4pr. In this article, we determine Rm(k,H4pr) and show that it is a subgroup of Cl(k). In particular, Rm(k,H4pr) is the full group Cl(k) if p does not divide the class number of k, or if either k contains a primitive p r th root of unity or p is unramified in k.

Local nearrings with generalized quaternion multiplicative group

A not necessarily zero-symmetric nearring R with identity is called local if the set of all non-invertible elements of R forms a subgroup of its additive group. The local nearrings whose multiplicative group is generalized quaternion are described. In particular, it is proved that their additive groups are abelian of types (3,3), (2,2,2,2), (2,2,4), (2,2,2,2,2) and (2,2,2,4). Keywords: Local nearring, Multiplicative group, Generalized quaternion group, Factorized group Mathematics Subject Classification (2000): 16N20, 16U60, 20M25

The quaternion group as a subgroup of the sphere braid groups

Let n be greater than or equal to 3. We prove that the quaternion group of order 8 is realised as a subgroup of the sphere braid group B\_n(S^2) if and only if n is even. If n is divisible by 4 then the commutator subgroup of B\_n(S^2) contains such a subgroup. Further, for all n greater than or equal to 3, B\_n(S^2) contains a subgroup isomorphic to the dicyclic group of order 4n.

Natural Geometry of Nonzero Quaternions

It is shown that the group of nonzero quaternions carries a family of natural closed Friedmann-Lemaitre-Robertson-Walker metrics.

Special values of L-functions by a Siegel–Weil–Kudla–Rallis formula

We study the arithmeticity of special values of L-functions attached to cuspforms which are Hecke eigenfunctions on hermitian quaternion groups Sp ∗ ( m , 0 ) which form a reductive dual pair with G = O ∗ ( 4 n ) . For f 1 and f 2 two cuspforms on H, consider their theta liftings θ f 1 and θ f 2 on G. Then we compute a Rankin–Selberg type integral and obtain an integral representation of the standard L-function: 〈 θ f 1 ⋅ E s , θ f 2 〉 G = 〈 f 1 , f 2 〉 H ⋅ L std ( f 1 , s ) . Also a short proof the Siegel–Weil–Kudla–Rallis formula is given. This implies that at the critical point s = s 0 = m − n + 1 2 Eisenstein series E s have rational Fourier coefficients. Via the natural embedding G × G ↪ G ˜ = O ∗ ( 8 n ) we restrict the holomorphic Siegel-type Eisenstein series E ˜ on G and decompose as a sum over an orthogonal basis for holomorphic cusp forms of fixed type. As a consequence we prove that the space of holomorphic cuspforms for O ∗ ( 4 n ) of given type is spanned by cuspforms so that the finite-prime parts of Fourier coefficients are rational and obtain special value results for the L-functions.

Three letters to Walter Feit on group representations and quaternions

Representations of the quaternion group by 2 × 2 matrices with coefficients in the ring of integers of an imaginary quadratic field.

TRIVIAL TORSION UNITS IN G-ADAPTED GROUP RINGS

Let G be a torsion group and let R be a G-adapted ring. In this note we study the question of when the group ring RG has only trivial torsion units. It turns out that the above question is closely related to the question of when the quaternion group ring RQ8 has only trivial torsion units. We first give a ring-theoretic condition on R which determines exactly when the quaternion group ring has only trivial torsion units. Then several equivalent conditions for RG to have only trivial torsion units are provided. We also investigate the hypercenter of the unit group of a G-adapted group ring RG, and show that when R satisfies the torsion trivial involution condition, this hypercenter is not equal to the center if and only if G is a Q*-group.

Hochschild cohomology of algebras of quaternion type, I: Generalized quaternion groups

Hochschild cohomology of algebras of quaternion type, I: Generalized quaternion groups

Hochschild Cohomology Ring of the Mod-2 Group Ring of the Generalized Quaternion Group

We will determine the ring structure of the Hochschild cohomology HH*(𝔽 2 Q t ) of the mod-2 group ring 𝔽 2 Q t for arbitrary generalized quaternion groups Q t of order 4t by calculating the ordinary cup product in H*(Q t , ψ 𝔽 2 Q t ).

Quaternionic representation of the Coxeter groupW(H4) and the polyhedra

The vertices of the four-dimensional polytope {3, 3, 5} and its dual {5, 3, 3} admitting the symmetry of the non-crystallographic Coxeter group W(H4) of order 14 400 are represented in terms of quaternions with unit norm where the polytope {3, 3, 5} is represented by the elements of the binaryicosahedral group of quaternions of order 120. We projected the polytopes to three-dimensional Euclidean space where the quaternionic vertices are the orbits of the Coxeter group W(H3), icosahedral group with inversion, where W(H3) × Z2 is one of the maximal subgroups of the Coxeter group W(H4). The orbits of the icosahedral group W(H3) in the polytope {3, 3, 5} are the conjugacy classes of the binary icosahedral group and represent a number of icosahedrons, dodecahedrons and one icosidodecahedron in three dimensions. The 15 orbits of the icosahedral group W(H3) in the polytope {5, 3, 3} represent the dodecahedrons, icosidodecahedrons, small rhombicosidodecahedrons and some convex solids possessing the icosahedral symmetry. One of the convex solids with 60 vertices is very similar to the truncated icosahedron (soccer ball) but with two different edge lengths which can be taken as a realistic model of the C60 molecule at extreme temperature and pressure.