The narrow class groups of the ℤ<Subscript>17</Subscript>- and ℤ<Subscript>19</Subscript>-extensions over the rational field

Let p be either 17 or 19, let ℤ _p denote the ring of p-adic integers, and let l be a prime number which is a primitive root modulo p ². We shall prove, with the help of a computer, that the l-class group of the ℤ _p -extension over the rational field is trivial. We shall also prove the triviality of the narrow 2-class group of the same ℤ _p -extension.     

On ℑ<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-normal subgroups of finite groups

Given a class ℑ of finite groups, a subgroup H of a group G is called ℑ _n -normal in G if there exists a normal subgroup T of G such that HT is a normal subgroup of G and (H ∩ T)H _G /H _G is contained in the ℑ-hypercenter Z _∞^ℑ (G/H _G ) of G/H _G . We obtain some results about the ℑ _n -normal subgroups and use them to study the structure of some groups.     

Simple ℤ<Subscript>2</Subscript>-actions twisted by aperiodic automorphisms

We construct a simple transformation whose centralizer is isomorphic to (?)² ? _A ?, where A is an aperiodic automorphism of ?². As a corollary we obtain an example of a ?²-action \({({T_g})_{g \in {Z^2}}}\) with minimal self-joinings which is conjugate to the (twisted by A) action \({({T_{Ag}})_{g \in {Z^2}}}\). This answers a question of E. Glasner.     

Dyadic Diaphony of Digital Nets Over ℤ<Subscript>2</Subscript>

The dyadic diaphony, introduced by Hellekalek and Leeb, is a quantitative measure for the irregularity of distribution of point sets in the unit-cube. In this paper we study the dyadic diaphony of digital nets over ℤ₂. We prove an upper bound for the dyadic diaphony of nets and show that the convergence order is best possible. This follows from a relation between the dyadic diaphony and the discrepancy. In order to investigate the case where the number of points is small compared to the dimension we introduce the limiting dyadic diaphony, which is defined as the limiting case where the dimension tends to infinity. We obtain a tight upper and lower bound and we compare this result with the limiting dyadic diaphony of a random sample.The first author is supported by the Australian Research Council under its Center of Excellence Program.The second author is supported by the Austrian Research Foundation (FWF), Project S 8305 and Project P17022-N12.     

A volume formula for ℤ<Subscript>2</Subscript>-symmetric spherical tetrahedra

We obtain formulas for the volume of a spherical tetrahedron with ℤ₂-symmetry realized as rotation about the axis passing through the midpoints of a pair of skew edges. We show the dependence of the volume formula on the edge lengths and dihedral angles of the tetrahedron. Several different formulas result whose scopes are determined by the geometric characteristics of the tetrahedron.     

Siegmund Duality for Continuous Time Markov Chains on ℤ<Stack><Subscript>+</Subscript><Superscript>d</Superscript></Stack>

For the continuous time Markov chain with transition function P(t) on Z₊^d, we give the necessary and sufficient conditions for the existence of its Siegmund dual with transition function P(t). If Q, the q-matrix of P(t), is uniformly bounded, we show that the Siegmund dual relation can be expressed directly in terms of q-matrices, and a sufficient condition under which the Q-function is the Siegmund dual of some Q-function is also given.     

Irreducible ℤ<Subscript>+</Subscript>-modules of near-group fusion ring <Emphasis Type="Italic">K</Emphasis>(ℤ<Subscript>3</Subscript>, 3)

The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible ?₊-modules over the near-group fusion ring K(?₃, 3) are explicitly classified. It turns out that there are only four inequivalent irreducible ?₊-modules of rank 2 and two inequivalent irreducible ?₊-modules of rank 4 over K(?₃, 3).     

On ℤ<Subscript>2</Subscript>-graded identities of the super tensor product of <Emphasis Type="Italic">UT</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">F</Emphasis>) by the Grassmann algebra

Let F be a field of characteristic zero and E be the unitary Grassmann algebra generated over an infinite-dimensional F-vector space L. Denote by \(\mathcal{E} = \mathcal{E}^{(0)} \oplus \mathcal{E}^{(1)}\) an arbitrary ?₂-grading of E such that the subspace L is homogeneous. Given a superalgebra A = A ⁽⁰⁾ ⊕ A ⁽¹⁾, define the superalgebra \(A\hat \otimes \mathcal{E}\) by \(A\hat \otimes \mathcal{E} = (A^{(0)} \otimes \mathcal{E}^{(0)} ) \oplus (A^{(1)} \otimes \mathcal{E}^{(1)} )\). Note that when E is the canonical grading of E then \(A\hat \otimes \mathcal{E}\) is the Grassmann envelope of A. In this work we find bases of ?₂-graded identities and we describe the ?₂-graded codimension and cocharacter sequences for the superalgebras \(UT_2 (F)\hat \otimes \mathcal{E}\), when the algebra UT ₂(F) of 2 ×2 upper triangular matrices over F is endowed with its canonical grading.     

The <Emphasis Type="Italic">D</Emphasis><Subscript>π</Subscript> property of finite groups in the case 2 ∉ π

The characterization of finite simple groups with the D _π property for any set π of odd prime numbers is completed. It was proved earlier that a finite group has the D _π property if and only if each of its composition factors has this property, hence the results of the paper provide an exhaustive characterization of the D _π property for all finite groups with known composition factors in the case 2 ? π.     

A Ring Theoretic Construction of Hadamard Difference Sets in ℤ<Subscript>8</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×ℤ<Subscript>2</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let be the Galois ring of characteristic 2³ and rank n and let . We give an explicit construction of Hadamard difference sets in .}Research supported by NSA grant MDA 904-02-1-0080.     

Hyperbolic Chevalley groups on ℂ<Superscript>2</Superscript>

Let Γ ? U (1, 1) be the subgroup generated by the complex reflections. Suppose that Γ acts discretely on the domain K = {(z ₁, z ₂) ∈ ?² ||z ₁|² ? |z ₂|² < 0} and that the projective group PΓ acts on the unit disk B = {|z ₁/z ₂| < 1} as a Fuchsian group of signature (n ₁, ..., n _s ), s ? 3, n _i ? 2. For such groups, we prove a Chevalley type theorem, i.e., find a necessary and sufficient condition for the quotient space K/Γ to be isomorphic to ?² ? {0}.     

On complexity reduction of Σ<Subscript>1</Subscript> formulas

 For a fixed q  ℕ and a given Σ₁ definition φ(d,x), where d is a parameter, we construct a model M of 1 Δ₀ + ? exp and a non standard d  M such that in M either φ has no witness smaller than d or phgr; is equivalent to a formula ϕ(d,x) having no more than q alternations of blocks of quantifiers. Received: 29 September 1998 / Revised version: 7 November 2001 Published online: 10 October 2002 RID="⋆" ID="⋆" Research supported in part by The State Committee for Scientific Research (Poland), KBN, grant number 2 PO3A 018 13. RID="⋆" ID="⋆" Research supported in part by The State Committee for Scientific Research (Poland), KBN, grant number 2 PO3A 018 13.     

On groups acting by cohomogeneity one on the euclidean space ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

The paper studies closed Lie subgroups G ⊂ Iso(ℝ ⁿ ) acting by cohomogeneity one on ℝ ⁿ and prove that when there is no singular orbit, then there is a simply connected, solvable and closed Lie subgroup F ⊂ G which acts by cohomogeneity one on ℝ ⁿ and the two actions are orbit equivalent.     

Linear forms and quadratic uniformity for functions on ℤ<Subscript><Emphasis Type="Italic">N</Emphasis></Subscript>

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW10a] and [GW10b], [GW10c], we gave a complete answer to this question for groups of the form G = F _p ⁿ , provided p is not too small. In ℤ _N , substantial extra difficulties arise, of which the most important is that an “inverse theorem” even for the uniformity norm || ·||_U³{\left\| \cdot \right\|_{{U^3}}} requires a more sophisticated “local” formulation. When N is prime, ℤ _N is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW10a]. Moreover, we obtain a doubly exponential bound.     

Spin structures and spectra of ℤ<Subscript>2</Subscript><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>-manifolds

We give necessary and sufficient conditions for the existence of pin^± and spin structures on Riemannian manifolds with holonomy group ₂^k. For any n4 (resp. n6) we give examples of pairs of compact manifolds (resp. compact orientable manifolds) M₁, M₂, non homeomorphic to each other, that are Laplace isospectral on functions and on p-forms for any p and such that M₁ admits a pin^± (resp. spin) structure whereas M₂ does not.Mathematics Subject Classification (2000):58J53, 57R15, 20H15Partially supported by Conicet and grants from SecytUNC, Foncyt and AgCba.     

On <Emphasis Type="Italic">π</Emphasis>-S-permutable subgroups of finite groups

Let π be a set of primes. A subgroup H of a finite group G is said to be π-S-permutable in G if H permutes with every Sylow q-subgroup of G for all primes q ∈ π. The main aim of this paper is to establish structural results about the normal closure of π-S-permutable subgroups and p-subgroups permuting with all p′-subgroups for a single prime p. Our results stem from a recent article by Isaacs [5] and subsequent discussions with the authors about it.     

On the measurability of the conic sections’ family in the projective space<Emphasis FontCategory="NonProportional">P</Emphasis><Subscript>3</Subscript><Subscript>3</Subscript>

In this paper we prove that the non degenerate conic sections’ family in the projective spaceP ₃ is measurable.     

On estimates for the Fourier-Bessel integral transform in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(ℝ<Subscript>+</Subscript>)

Two estimates useful in applications are proved for the Fourier-Bessel integral transform in L ₂(?₊) as applied to some classes of functions characterized by a generalized modulus of continuity.     

Moduli spaces ℳ<Subscript>2,1</Subscript> and ℳ<Subscript>3,1</Subscript>

The cell structure of the spaces ℳ_2,1 and ℳ_3,1 is considered. These are the spaces of complex curves of genus 2 and 3 with one marked point. For the space ℳ_2,1, nine cells of the highest dimension 8 are described and their adjacency is studied. For the space ℳ_3,1, a list of all 1726 cells of the highest dimension 14 (with orientation) is obtained. The list of adjacent couples of cells is also obtained. These lists can be found on the web.     

Discrete subgroups of <Emphasis Type="Italic">PU</Emphasis> (2, 1) acting on P<Stack><Subscript>ℂ</Subscript><Superscript>2</Superscript></Stack> and the Kobayashi metric

In this paper, we prove following: If G ⊂ PU (2, 1) is an infinite, discrete group, acting on P_ℂ² without complex invariant lines, then the component containing ℍP_ℂ² of the domain of discontinuity Ω(G) = PP_ℂ²∖ Λ (G), according to Kulkarni, is G-invariant complete Kobayashi hyperbolic. The authors were supported by the Universidad Autónoma de Yucatán and the Universidad Nacional Autónoma de México.