On r*-sequenceability and symmetric harmoniousness of groups. II

In this paper we investigate symmetric harmoniousness of groups and connections of this concept to the R*-sequenceability of groups. We prove that, under suitable assumptions, the direct product of a symmetric harmonious group with a group that is R*-sequenceable is R*-sequenceable; we discuss the symmetric harmoniousness of abelian and of nilpotent groups; we also prove that, for a fixed odd prime p, all but possibly finitely many of the nonabelian groups of order pq (q prime, q ≡ 1 (mod p)) are symmetric harmonious. © 1995 John Wiley & Sons, Inc.     

Constructions of large translation nets with nonabelian translation groups

In this paper the first infinite series of translation nets with nonabelian translation groups and a large number of parallel classes are constructed. For that purpose we investigate partial congruence partitions (PCPs) with at least one normal component.Two series correspond to partial congruence partitions containing one normal elementary abelian component. The construction results by using some basic facts about the first cohomology group of the translation group G regarded as an extension of the normal component which itself is a group of central translations.The other series correspond to partial congruence partitions containing two normal nonabelian components. The constructions are based on the well known automorphism method which leads to so-called splitting translation nets. By investigating the Suzuki groups Sz(q), the protective unitary groups PSU(3, q ²) and the Ree groups R(q) as doubly transitive permutation groups, we obtain examples of nonabelian groups admitting a large number of pairwise orthogonal fixed-point-free group automorphisms.     

A Class of Imprimitive Association Schemes

From Burnside's p^αq^β-Theorem, it follows that any nonabelian group of order p^αq^β, where p and q are primes, cannot be simple. As a main result of this article, we state and prove an analog of the mentioned theorem for commutative association schemes.     

On Composition Factors of Finite Groups Having the Same Set of Element Orders as the Group U3(q)

We study the composition factors of a finite nonsolvable group having the same set of order elements as the simple unitary group U ₃(q) for an odd q. We prove in particular that for q>5 the (only) nonabelian composition factor of such a group is isomorphic to U ₃(q).     

On Finite Nonabelian Loop Capable Groups

We say that groups, which are isomorphic to inner mapping groups of finite loops, are loop capable. Let p and q be distinct prime numbers, S a nonabelian group of order pq, and C a finite nontrivial cyclic group such that gcd (|S|, |C|) = 1. We show that the group S × C is not loop capable.     

ON NONABELIAN TENSOR PRODUCT MODULO q OF GROUPS

The nonabelian tensor product modulo q of two crossed modules of groups is investigated, where q is a positive integer. It is obtained a six term exact sequence of groups connecting the nonabelian tensor product modulo q with algebraic K-functor K ₂ with Z _q coefficients for (noncommutative) local rings. The notion of q-homology groups of a group G with coefficients in a G-module A is introduced, some its properties and calculations are given. The relationship between q-homology groups and derived functors of tensor product modulo q is studied.     

Weyl quantization of Lebesgue spaces

We study boundedness and compactness properties for the Weyl quantization with symbols in L^q (?^2d ) acting on L^p (?^d ). This is shown to be equivalent, in suitable Banach space setting, to that of the Wigner transform. We give a short proof by interpolation of Lieb's sufficient conditions for the boundedness of the Wigner transform, proving furthermore that these conditions are also necessary. This yields a complete characterization of boundedness for Weyl operators in L^p setting; compactness follows by approximation. We extend these results defining two scales of spaces, namely L_*^q (?^2d ) and L_?^q (R^2d ), respectively smaller and larger than the L^q (?^2d ),and showing that the Weyl correspondence is bounded on L_*^q (R^2d ) (and yields compact operators), whereas it is not on L_?^q (R^2d ). We conclude with a remark on weak‐type L^p boundedness (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the nonabelian tensor square and capability of groups of order <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript><Emphasis Type="Italic">q</Emphasis>

A group G is said to be capable if it is isomorphic to the central factor group H/Z(H) for some group H. Let G be a nonabelian group of order p ² q for distinct primes p and q. In this paper, we compute the nonabelian tensor square of the group G. It is also shown that G is capable if and only if either Z(G) = 1 or p < q and G^ab=\mathbbZ_p×\mathbbZ_p{G^{\rm ab}=\mathbb{Z}_{p}\times\mathbb{Z}_{p}} .     

Verifying Huppert’s Conjecture for <Superscript>2</Superscript><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H×A, where A is an abelian group. In this paper, we verify the conjecture for the twisted Ree groups ² G ₂(q ²) for q ² = 3^2m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many of the nonabelian simple groups.     

F*-Rings Are O*

O ^*-rings were introduced by Fuchs and recently characterized by Steinberg. A ring R is called O ^* if every partial order on R extends to a total order. We weaken the condition on the ordering of the ring by requiring that every partial order on R extends to an f-order. We call those rings F ^*-rings. We show that the two classes of rings coincide.     

Subgroups of GL 1(R) for Local Rings R

Abstract

Let R be a local ring, with maximal ideal m , and residue class division ring R/ m ?=?D. Denote by R*?=?G L ₁(R), the group of units of R. Here we investigate some algebraic structure of subnormal and maximal subgroups of R*. For instance, when D is of finite dimension over its center, it is shown that finitely generated subnormal subgroups of R* are central. It is also proved that maximal subgroups of R* are not finitely generated. Furthermore, assume that P is a nonabelian maximal subgroup of R* such that P contains a noncentral soluble normal subgroup of finite index, it is shown that D is a crossed product division algebra.     

Gallot–Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications

We extend the Gallot–Tanno theorem to closed pseudo-Riemannian manifolds. It is done by showing that if the cone over a manifold admits a parallel symmetric (0, 2)-tensor then it is Riemannian. Applications of this result to the existence of metrics with distinct Levi-Civita connections but having the same unparametrized geodesics and to the projective Obata conjecture are given. We also apply our result to show that the holonomy group of a closed (O(p + 1, q), S ^p,q )-manifold does not preserve any nondegenerate splitting of \mathbb R^p+1,q{\mathbb {R}^{p+1,q}}.     

Fejér means of functions on noncommutative Vilenkin groups with respect to the character system

Summary It is known that the Fejér means - with respect to the character system of the Walsh, and bounded Vilenkin groups - of an integrable function converge to the function a.e. In this work we discuss analogous problems on the complete product of finite, not necessarily Abelian groups with respect to the character system for functions that are constant on the conjugacy classes. We find that the nonabelian case differs from the commutative case. We prove the a.e. convergence of the (C,1) means of the Fourier series of square integrable functions. We also prove the existence of a function f∊L^q for some q >1 such that sup |σ_n f | = + ∞ a.e. This is a sharp contrast between the Abelian and the nonabelian cases.     

Some new symmetric designs

For q, an odd prime power, we construct symmetric (2q²+2q+1,q²,½q(q-1)) designs having an automorphism group of order q that fixes 2q+1 points. The construction indicates that for each q the number of such designs that are pairwise non-isomorphic is very large.     

A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (2^2d+2, 2^2d+1±2 ^d , 2^2d ±2 ^d ). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2 ^d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2^2d+2 with exponent 2 ^d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.     

On symmetric nets and generalized Hadamard matrices from affine designs

Symmetric nets are affine resolvable designs whose duals are also affine. It is shown that. up to isomorphism, there are exactly four symmetric (3, 3)-nets (v=b=27,k=9), and exactly two inequivalent 9×9 generalized Hadamard matrices over the group of order 3. The symmetric (3, 3)-nets are found as subnets of affine resolvable 2-(27, 9, 4) designs. Ten of the 68 non-isomorphic affine resolvable 2-(27, 9, 4) designs are not extensions of symmetric (3, 3)-subnets, providing the first examples of affine 2-(q³, q², q²–1/q–1) designs without symmetric (q, q)-subnets.     

Investigation of solvable (120, 35, 10) difference sets

Abelian difference sets with parameters (120, 35, 10) were ruled out by Turyn in 1965. Turyn's techniques do not apply to nonabelian groups. We attempt to determine the existence of (120, 35, 10) difference sets in the 44 nonabelian groups of order 120. We prove that if a solvable group admits a (120, 35, 10) difference set, then it admits a quotient group isomorphic to the cyclic group of order 24 or to U₂₄ ? 〈x,y : x⁸ = y³ = 1, xyx^?1 = y^?1〉. We describe a computer search, which rules out solutions with a ?₂₄ quotient. The existence question remains undecided in the three solvable groups admitting a U₂₄ quotient. The question also remains undecided for the three nonsolvable groups of order 120. © 2004 Wiley Periodicals, Inc.     

A unified construction for c*· c- and L · L*-geometries in projective spaces

We present two new constructions for c^* · c-geometries. The first provides, for each even prime powerq, a flag-transitive c^* · c-geometry of orderq–1 that is embedded in the projective space PG(3,q) and which is related with the Cameron-Fisher extended grids of odd type. The second construction is valid independently of the parity ofq. Forq even, it produces the same geometry as the first construction, and forq odd, two geometries related with some extended grids constructed by Meixner and Pasini.Next, by using some complementary models for c^* and L in a projective plane, we derive from our construction a new family of L · L^*-geometries embedded in PG(3,q). Forq even, these geometries are flag-transitive.     

On generalized Hadamard matrices of minimum rank

Generalized Hadamard matrices of order qⁿ⁻¹ (q—a prime power, n2) over GF(q) are related to symmetric nets in affine 2-(qⁿ,qⁿ⁻¹,(qⁿ⁻¹−1)/(q−1)) designs invariant under an elementary abelian group of order q acting semi-regularly on points and blocks. The rank of any such matrix over GF(q) is greater than or equal to n−1. It is proved that a matrix of minimum q-rank is unique up to a monomial equivalence, and the related symmetric net is a classical net in the n-dimensional affine geometry AG(n,q).     

Group algebras with symmetric units satisfying a group identity

We study group algebras FG for which the symmetric units under the natural involution: g^*=g⁻¹ satisfy a group identity. For infinite fields F of characteristic ≠2, a classification of torsion groups G whose symmetric units U⁺(FG) satisfy a group identity was given in [3] by Giambruno-Sehgal-Valenti. We extend this work to non torsion groups. Research supported by NSERC of Canada and MIUR of Italy.