On Suzuki's characterization ofPGL(2,2 f )

A character theoretic result of W. Feit asserts, that if G is a Zassenhaus group of odd degree n+1, then n=2^for some f>0. Using this result, M. Suzuki proved the following two results.Theorem 1 Let G be a Zassenhaus group of odd degree n+1. tThen G=PGL(2, 2^f) if and only if j=g^* in the structure equation of G.Theorem 2 Suppose that G is a Zassenhaus group of odd degree n+1 and of order (n+1)nd. If 3 divides d, then G=PGL(2, 2^f).We give an elementary proof of these theorems, avoiding the use of Feit's result.     

Product of a biprimary and A 2-decomposable group

Suppose a finite group G is the product of a subgroups A and B of coprime orders, and suppose the order of A is p ^a q^b, where p and q are primes, and B is a 2-decomposable group of even order. Assume that a Sylow p-subgroup P is cyclic. If the order of P is not equal to 3 or 7, then G is solvable. If G is nonsolvable and G contains no nonidentity solvable invariant subgroups, then G is isomorphic to PSL(2, 7) or PGL(2, 7).Translated from Matematicheskie Zametki, Vol. 23, No. 5, pp. 641–649, May, 1978.     

The structure of symmetry groups of GK(2n,G)

The automorphism groups of the one-factorizations GK(2n,G) are computed. It is shown that every 1-factorization of K_2n with a subgroup of the automorphism group that acts sharply 2-transitively on the one-factors must be GK(p^m + 1, (Z_p)^m) for some odd prime p. © 1994 John Wiley & Sons, Inc.     

On the symmetric square Unstable twisted characters

We provide a purely local computation of the (elliptic) twisted (by “transpose-inverse”) character of the representationπ=I(1) of PGL(3) over ap-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of [IV] of automorphic and admissible representations of SL(2) to PGL(3). It leads — see [FK] — to a proof of the (unstable) fundamental lemma in the theory of the symmetric square lifting, namely that corresponding spherical functions (on PGL(2) and PGL(3)) are matching: they have matching orbital integrals. The new case in [FK] is the unstable one. A direct local proof of the fundamental lemma is given in [V].     

A family of regular graphs of girth 5

Murty [A generalization of the Hoffman-Singleton graph, Ars Combin. 7 (1979) 191-193.] constructed a family of (p^m+2)-regular graphs of girth five and order 2p^2m, where p?5 is a prime, which includes the Hoffman-Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497-504]. This construction gives an upper bound for the least number f(k) of vertices of a k-regular graph with girth 5. In this paper, we extend the Murty construction to k-regular graphs with girth 5, for each k. In particular, we obtain new upper bounds for f(k), k?16.     

The geometrical structure in a generic generating space of a non-developable generalized ruled surface in the hyperbolic space Hm

A generalized ruled surface (G.R.S.) in H^m is generated by ¹ n-dimensional totally geodesic subspaces of H^m, which are called generators of the G.R.S. In this paper the basic results about points of striction, parameters of distribution and Riemann curvature at the points of a fixed generator of the G.R.S. are obtained, using a method which makes any representation of the G.R.S. in H^m superfluous.     

LENGTH ONE IDEAL EXTENSIONS AND THEIR ASSOCIATED GRADED RINGS

Let (R. m) be a d-dimensional Cohen-Macaulay local ring. Given m-primary ideals J ? I of R such that I is contained in the integral closure of J and λ(I/J)= I, we compare depth G(J) and depth G(J). For example, if J has reduction number one, JI = I², and μ(J)≤ d + 1, we prove that depth G(I)≥d – 1. If, in addition, μ(I)= d + 1, we show that I has reduction number one, and hence G(I) is Cohen-Macaulay. These results, besides leading to statements comparing depths of associated graded rings along a composition series, make visible the possibility of studying powers of an ideal by using reductions that are not minimal reductions.     

Optimal Extension of Fourier Multiplier Operators in L p (G)

Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier ${\psi : \Gamma \to {\mathbb C}}Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier y: G? \mathbb C{\psi : \Gamma \to {\mathbb C}} (with Γ the dual group), we study the optimal domain of the multiplier operator T^(p)_y : L^p (G) ? L^p (G){T^{(p)}_\psi : L^p (G) \to L^p (G)}. This is the largest Banach function space, denoted by L¹(m^(p)_y){L^1(m^{(p)}_\psi)}, with order continuous norm into which L ^p (G) is embedded and to which T^(p)_y{ T^{(p)}_\psi} has a continuous L ^p (G)-valued extension. Compactness conditions for the optimal extension are given, as well as criteria for those ψ for which L¹(m^(p)_y) = L^p (G){L^1(m^{(p)}_\psi) = L^p (G)} is as small as possible and also for those ψ for which L¹(m^(p)_y) = L¹ (G){L^1(m^{(p)}_\psi) = L^1 (G)} is as large as possible. Several results and examples are presented for cases when L^p (G) \subsetneqq L¹(m^(p)_y) \subsetneqq L¹ (G){L^p (G) \subsetneqq L^1(m^{(p)}_\psi) \subsetneqq L^1 (G)}.     

Infinite families of 2- and 3-designs with parameters v = p + 1, k = (p − 1)/2i + 1, where p odd prime, 2e T (p − 1), e ≥ 2, 1 ≤ i ≤ e

Let p be an odd prime number such that p − 1 = 2^em for some odd m and e ≥ 2. In this article, by using the special linear fractional group PSL(2, p), for each i, 1 ≤ i ≤ e, except particular cases, we construct a 2-design with parameters v = p + 1, k = (p − 1)/2ⁱ + 1 and λ = ((p − 1)/2ⁱ+1)(p − 1)/2 = k(p − 1)/2, and in the case i = e we show that some of these 2-designs are 3-designs. Likewise, by using the linear fractional group PGL(2,p) we construct an infinite family of 3-designs with the same v k and λ = k(k − 2). These supplement a part of [4], in which we gave an infinite family of 3-designs with parameters v = q + 1, k = (q + 1)/2 = (q − 1)/2 + 1 and λ = (q + 1)(q − 3)/8 = k(k − 2)/2, where q is a prime power such that q − 1 = 2m for some odd m and q > 7. Some of the designs given in this article and in [4] fill in a few blanks in the table of Chee, Colbourn, and Kreher [2]. © 1997 John Wiley & Sons, Inc.     

Orthogonal basis for spherical monogenics by step two branching

Spherical monogenics can be regarded as a basic tool for the study of harmonic analysis of the Dirac operator in Euclidean space \mathbb R^m{{\mathbb R}^m}. They play a similar role as spherical harmonics do in case of harmonic analysis of the Laplace operator on \mathbb R^m{{\mathbb R}^m}. Fix the direct sum \mathbb R^m=\mathbb R^p ?\mathbb R^q{{\mathbb R}^m={\mathbb R}^p \oplus {\mathbb R}^q}. In this article, we will study the decomposition of the space M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)} of spherical monogenics of order n under the action of Spin(p) × Spin(q). As a result, we obtain a Spin(p) × Spin(q)-invariant orthonormal basis for M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}. In particular, using the construction with p = 2 inductively, this yields a new orthonormal basis for the space M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}.     

On divisible designs and local algebras

We study the action of the group PGL(m,A) on the projective space PG(m − 1,A) over a finite commutative local algebra A in order to construct a class of divisible designs, denoted by D_m(d,A), which is the classical one of 2-designs (of points and of flats of fixed projective dimension) in the case where A is a field. We also study the constructed divisible designs with particular care for the case where d = m − 1. © 1995 John Wiley & Sons, Inc.     

A characterization of PGL(2, q), q odd

Following the lines of [10], we give a characterization of the group PGL(2, q), q odd, in terms of involutions.Work performed under the auspicies of G.N.S.A.C.A. of C.N.R. supported by the 40% grants of M.P.I.     

Quantization causes waves: Smooth finitely computable functions are affine

Every automaton (a letter-to-letter transducer) A whose both input and output alphabets are F _p = {0, 1,..., p - 1} produces a 1-Lipschitz map f _A from the space Z _p of p-adic integers to Z _p . The map fA can naturally be plotted in a unit real square I² ? R²: To an m-letter non-empty word v = γ _m-1γ _m-2... γ0 there corresponds a number 0.v ∈ R with base-p expansion 0.γ _m-1γ _m-2... γ0; so to every m-letter input word w = α _m-1α _m-2 ··· α0 of A and to the respective m-letter output word a(w) = β _m-1β _m-2 ··· β0 of A there corresponds a point (0.w; 0.a(w)) ∈ R². Denote P(A) a closure of the point set (0.w; 0.a(w)) where w ranges over all non-empty words.We prove that once some points of P(A) constitute a C ²-smooth curve in R², the curve is a segment of a straight line with a rational slope. Moreover, when identifying P(A) with a subset of a 2-dimensional torus T² ∈ R³, the smooth curves from P(A) constitute a collection of torus windings which can be ascribed to complex-valued functions ψ(x, t) = e ^i(Ax-2πBt) (x, t ∈ R), i.e., to matter waves. As automata are causal discrete systems, the main result may serve a mathematical reasoning why wave phenomena are inherent in quantum systems: This is just because of causality principle and discreteness of matter.     

Self-concordant barriers for hyperbolic means

The geometric mean and the function (det(·)) ^1/m (on the m-by-m positive definite matrices) are examples of “hyperbolic means”: functions of the form p ^1/m , where p is a hyperbolic polynomial of degree m. (A homogeneous polynomial p is “hyperbolic” with respect to a vector d if the polynomial t↦p(x+td) has only real roots for every vector x.) Any hyperbolic mean is positively homogeneous and concave (on a suitable domain): we present a self-concordant barrier for its hypograph, with barrier parameter O(m ²). Our approach is direct, and shows, for example, that the function −mlog(det(·)−1) is an m ²-self-concordant barrier on a natural domain. Such barriers suggest novel interior point approaches to convex programs involving hyperbolic means. Received: December 2, 1999 / Accepted: February 2001?Published online September 3, 2001     

An inversion formula for the dual horocyclic Radon transform on the hyperbolic plane

Consider the Poincare unit disk model for the hyperbolic plane H ². Let Ξ be the set of all horocycles in H ² parametrized by (θ, p), where e^iθ is the point where a horocycle ξ is tangent to the boundary |z| = 1, and p is the hyperbolic distance from ξ to the origin. In this paper we invert the dual Radon transform R* : μ(θ, p) → (z) under the assumption of exponential decay of μ and some of its derivatives. The additional assumption is that P_m(d/dp)(μ_m(p)e^p) be even for all m ∈ ?. Here P_m(d/dp) is a family of differential operators introduced by Helgason, and μ_m(p) are the coefficients of the Fourier series expansion of μ(θ, p). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Recognition of finite simple groupsL 3(2 m ) ANDU 3(2 m ) by their element orders

It is proved that a finite group isomorphic to a simple non-Abelian group L₃(2^m) or U₃(2^m) is, up to isomorphism, recognizable by a set of its element orders. On the other hand, for every simple group S=S₄(2^m), there exist infinitely many pairwise non-isomorphic groups G with w(G)=w(S). As a consequence, we present a list of all recognizable finite simple groups G, for which 4t ∉ ω(G) with t>1. Supported by RFFR grant No. 99-01-00550, by the National Natural Science Foundation of China (grant No. 19871066), and by the State Education Ministry of China (grant No. 98083). Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 567–585, September–October, 2000.     

Polyhedral groups and graph amalgamation products

A polyhedral group G is defined to be the orientation-preserving subgroup of a discrete reflection group acting on hyperbolic 3-space $\text{H}$³, and having a fundamental polyhedron of finite volume. A special presentation for G is obtained from the geometry of the polyhedron. This gives G the structure of a graph amalgamation product, and which, in some cases, splits as a free product with amalgamation. The simplest examples of polyhedral groups are the so-called tetrahedral groups. Other examples are given amongst the the groups PGL(2,O_m), where O_m is the ring of algebraic integers in the quadratic imaginary field $\text{Q(}\text{-m}\text{)}$, m>0.     

A class of ((p n−p m)(p n−1), p n−p m)-arcs in PG(2, p n)

A (k, d)-arc in PG(2, q) is a set of k points such that some d, but no d+1, of them are collinear. An outstanding problem is to find the maximum value of k for which a (k, d)-arc exists. A construction is given for a class of (k, p ⁿ–p ^m)-arcs in PG(2, p ⁿ). These arcs constitute a lower bound on the maximum possible value of k, and a subset of them is shown to be optimal.     

Sharply 1-transitive subsets of certain permutation groups

In this paper we give a method for constructing sharply 1-transitive permutation sets inside a finite permutation group with certain properties and we apply this method to obtain a family of sharply 1-transitive permutation subsets of the sharply 3-transitive permutation group M(p ^2f ) on PG(1, p ^2f ) for p ^f 1 (mod 4).Work supported by G.N.S.A.G.A. and M.P.I.     

The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods

A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x² = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of K_n if n + 1 = p^q for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring.