Integral group ring of the Suzuki sporadic simple group

Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Suzuki sporadic simple group Suz. As a consequence, for this group we confirm the Kimmerle`s conjecture on prime graphs.

INVOLUTIONS AND FREE PAIRS OF BASS CYCLIC UNITS IN INTEGRAL GROUP RINGS

Let ZG be the integral group ring of the finite nonabelian group G over the ring of integers Z, and let * be an involution of ZG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (u(k,m)(x*), u(k,m)(x*)) or (u(k,m)(x), u(k,m)(y)), formed respectively by Bass cyclic and *-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ZG.

CENTRAL UNITS IN METACYCLIC INTEGRAL GROUP RINGS

In this article, we give a method to compute the rank of the subgroup of central units of ZG, for a finite metacyclic group, G, by means of Q-classes and R-classes. Then we construct a multiplicatively independent set u subset of Z(U(ZC(p,q))) and by applying our results, we prove that u generates a subgroup of finite index.

BASS CYCLIC UNITS AS FACTORS IN A FREE GROUP IN INTEGRAL GROUP RING UNITS

Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ZG of a solvable and finite group G, such that u has infinite order modulo the center of U(ZG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ZG which is either a Bass cyclic unit or a bicyclic unit.

ALTERNATING UNITS AS FREE FACTORS IN THE GROUP OF UNITS OF INTEGRAL GROUP RINGS

Let G be a group of odd order that contains a non-central element x whose order is either a prime p >= 5 or 3(l), with l >= 2. Then, in U(ZG), the group of units of ZG, we can find an alternating unit u based on x, and another unit v, which can be either a bicyclic or an alternating unit, such that for all sufficiently large integers m we have that < u(m), v(m)> = < u(m)> * < v(m)> congruent to Z * Z.

Multiple Brake Orbits and Homoclinics in Riemannian Manifolds

Let (M, g) be a complete Riemannian manifold, Omega subset of Man open subset whose closure is homeomorphic to an annulus. We prove that if a,Omega is smooth and it satisfies a strong concavity assumption, then there are at least two distinct geodesics in starting orthogonally to one connected component of a,Omega and arriving orthogonally onto the other one. Using the results given in Giamb et al. (Adv Differ Equ 10:931-960, 2005), we then obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. Under a further symmetry assumption, the result is improved by showing the existence of at least dim(M) pairs of geometrically distinct geodesics as above, brake orbits and homoclinic orbits. In our proof we shall use recent deformation results proved in Giamb et al. (Nonlinear Anal Ser A: Theory Methods Appl 73:290-337, 2010).

Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.

Involutions and free pairs of bicyclic units in integral group rings

If * : G -> G is an involution on the finite group G, then * extends to an involution on the integral group ring Z[G] . In this paper, we consider whether bicyclic units u is an element of Z[G] exist with the property that the group < u, u*> generated by u and u* is free on the two generators. If this occurs, we say that (u, u*)is a free bicyclic pair. It turns out that the existence of u depends strongly upon the structure of G and on the nature of the involution. One positive result here is that if G is a nonabelian group with all Sylow subgroups abelian, then for any involution *, Z[G] contains a free bicyclic pair.

Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and homeomorphic to the N-dimensional disk

In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link (proved in Giambo et al. (2005) [8]) of the multiplicity problem with the famous Seifert conjecture (formulated in Seifert (1948) [1]) about multiple brake orbits for a class of Hamiltonian systems at a fixed energy level. (C) 2010 Elsevier Ltd. All rights reserved.

Birkhoffian Systems in Infinite Dimensional Manifolds

We generalize the theory of Kobayashi and Oliva (On the Birkhoff Approach to Classical Mechanics. Resenhas do Instituto de Matematica e Estatistica da Universidade de Sao Paulo, 2003) to infinite dimensional Banach manifolds with a view towards applications in partial differential equations.

Associated Family of G-Structure Preserving Minimal Immersions in Semi-Riemannian Manifolds

We prove the existence of an associated family of G-structure preserving minimal immersions into semi-Riemannian manifolds endowed with a compatible infinitesimally homogeneous G-structure. We will study in more details minimal embeddings into product of space forms.