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.

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.

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.

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.

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.

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.

Bicyclic units, Bass cyclic units and free groups

Let G be a finite group and ZG its integral group ring. We show that if alpha is a nontrivial bicyclic unit of ZG, then there are bicyclic units beta and gamma of different types, such that and are non-abelian free groups. In the case when G is non-abelian of order coprime to 6 we prove the existence of a bicyclic unit u and a Bass cyclic unit v in ZG, such that < u(m), v > is a free non-abelian group for all sufficiently large positive integers m.

Cellular decomposition of quaternionic spherical space forms

We obtain an explicit cellular decomposition of the quaternionic spherical space forms, manifolds of positive constant curvature that are factors of an odd sphere by a free orthogonal action of a generalized quaternionic group. The cellular structure gives and explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group. As an application we compute the Whitehead torsion of these spaces for any representation of the fundamental group. © 2012 Springer Science+Business Media B.V.

Hypercentral unit groups and the hyperbolicity of a modular group algebra

We classify groups G such that the unit group U-1 (ZG) is hypercentral. In the second part, we classify groups G whose modular group algebra has hyperbolic unit groups U-1 (KG).

Computations in Relative Algebraic K-Groups

Let G be finite group and K a number field or a p-adic field with ring of integers O_K. In the first part of the manuscript we present an algorithm that computes the relative algebraic K-group K_0(O_K[G],K) as an abstract abelian group. We solve the discrete logarithm problem, both in K_0(O_K[G],K) and the locally free class group cl(O_K[G]). All algorithms have been implemented in MAGMA for the case K = \IQ. In the second part of the manuscript we prove formulae for the torsion subgroup of K_0(\IZ[G],\IQ) for large classes of dihedral and quaternion groups.

ANTISYMMETRIC ELEMENTS IN GROUP RINGS II

Let R be a commutative ring, G a group and RG its group ring. Let phi : RG -> RG denote the R-linear extension of an involution phi defined on G. An element x in RG is said to be phi-antisymmetric if phi(x) = -x. A characterization is given of when the phi-antisymmetric elements of RG commute. This is a completion of earlier work.

On spectra of abelian group rings

In this paper we study the spectrum of integral group rings of finitely generated abelian groups G from the scheme-theoretic viewpoint. We prove that the (closed) singular points of Spec Z[G], the (closed) intersection points of the irreducible components of Spec Z[G] and the (closed) points over the prime divisors of vertical bar t(G)vertical bar coincide. We also determine the formal completion of Spec Z[G] at a singular point.

HYPERBOLICITY OF SEMIGROUP ALGEBRAS II

In 1996, Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki-Juriaans-Souza Filho continued this line of research by partially classifying the finite semigroups whose rational semigroup algebra contains a Z-order with hyperbolic unit group. In this paper, we complete this classification and give an easy proof that deals with all finite semigroups.

Sylow P-Subgroups of Abelian Group Rings

2000 Mathematics Subject Classification: Primary 20C07, 20K10, 20K20, 20K21; Secondary 16U60, 16S34.

The synthesis and application of novel profluorescent nitroxides as probes for polymer degradation

This PhD project has expanded the knowledge in the area of profluorescent nitroxides with regard to the synthesis and characterisations of novel profluorescent nitroxide probes as well as physical characterisation of the probe molecules in various polymer/physical environments. The synthesis of the first example of an azaphenalene-based fused aromatic nitroxide TMAO, [1,1,3,3-tetramethyl-2,3-dihydro-2-azaphenalen-2-yloxyl, was described. This novel nitroxide possesses some of the structural rigidity of the isoindoline class of nitroxides, as well as some properties akin to TEMPO nitroxides. Additionally, the integral aromatic ring imparts fluorescence that is switched on by radical scavenging reactions of the nitroxide, which makes it a sensitive probe for polymer degradation. In addition to the parent TMAO, 5 other azaphenalene derivatives were successfully synthesised. This new class of nitroxide was expected to have interesting redox properties when the structure was investigated by high-level ab initio molecular orbitals theory. This was expected to have implications with biological relevance as the calculated redox potentials for the azaphenalene ring class would make them potent antioxidant compounds. The redox potentials of 25 cyclic nitroxides from four different structural classes (pyrroline, piperidine, isoindoline and azaphenalene) were determined by cyclic voltammetry in acetonitrile. It was shown that potentials related to the one electron processes of the nitroxide were influenced by the type of ring system, ring substituents or groups surrounding the moiety. Favourable comparisons were found between theoretical and experimental potentials for pyrroline, piperidine and isoindoline ring classes. Substitution of these ring classes, were correctly calculated to have a small yet predictable effect on the potentials. The redox potentials of the azaphenalene ring class were underestimated by the calculations in all cases by at least a factor of two. This is believed to be due to another process influencing the redox potentials of the azaphenalene ring class which is not taken into account by the theoretical model. It was also possible to demonstrate the use of both azaphenalene and isoindoline nitroxides as additives for monitoring radical mediated damage that occurs in polypropylene as well as in more commercially relevant polyester resins. Polymer sample doped with nitroxide were exposed to both thermo-and photo-oxidative conditions with all nitroxides showing a protective effect. It was found that isoindoline nitroxides were able to indicate radical formation in polypropylene aged at elevated temperatures via fluorescence build-up. The azaphenalene nitroxide TMAO showed no such build-up of fluorescence. This was believed to be due to the more labile bond between the nitroxide and macromolecule and the protection may occur through a classical Denisov cycle, as is expected for commercially available HAS units. Finally, A new profluorescent dinitroxide, BTMIOA (9,10-bis(1,1,3,3- tetramethylisoindolin-2-yloxyl-5-yl)anthracene), was synthesised and shown to be a powerful probe for detecting changes during the initial stages of thermo-oxidative degradation of polypropylene. This probe, which contains a 9,10-diphenylanthracene core linked to two nitroxides, possesses strongly suppressed fluorescence due to quenching by the two nitroxide groups. This molecule also showed the greatest protective effect on thermo-oxidativly aged polypropylene. Most importantly, BTMIOA was found to be a valuable tool for imaging and mapping free-radical generation in polypropylene using fluorescence microscopy.