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.

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.

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.

On Groups Whose Maximal Cyclic Subgroups Are Maximal

We consider the problem of classifying those groups whose maximal cyclic subgroups are maximal. We give a complete classification of those groups with this property and which are either soluble or residually finite.

Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane

We classify the ( finite and infinite) virtually cyclic subgroups of the pure braid groups P(n)(RP(2)) of the projective plane. The maximal finite subgroups of P(n)(RP(2)) are isomorphic to the quaternion group of order 8 if n = 3, and to Z(4) if n >= 4. Further, for all n >= 3, the following groups are, up to isomorphism, the infinite virtually cyclic subgroups of P(n)(RP(2)): Z, Z(2) x Z and the amalgamated product Z(4)*(Z2)Z(4).

Stereoselective Nucleophilic Addition of Potassium Alkyltrifluoroborates to Cyclic N-Acyliminium Ions: a Simple and Mild Approach to Chiral 5-Alkyl-pyrrolidin-2-ones

The stereoselective nucleophilic addition of potassium alkyltrifluoroborates to cyclic N-acyliminium ions derived from N-benzyl-3,4,5-triacetoxy-pyrrolidin-2-one, which affords 5-substituted-pyrrolidin-2-ones, is described. The products are obtained in moderate to good yields and are produced predominantly as the anti diastereomer.

Deposition of selenium thin layers on gold surfaces from sulphuric acid media: Studies using electrochemical quartz crystal microbalance, cyclic voltammetry and AFM

In this paper we report here new considerations about the relationship between the mass and charge variations (m/z relationship) in underpotential deposition (UPD), bulk deposition and also in the H(2)Se formation reaction. Nanogravimetric experiments were able to show the adsorption of H(2)SeO(3) on the AuO surface prior to the voltammetric sweep and that, after the AuO reduction, 0.40 monolayer of H(2)SeO(3) remains adsorbed on the newly reduced Au surface, which was enough to gives rise to the UPD layer. The UPD results indicate that the maximum coverage with Se(ads) on polycrystalline gold surface corresponds to approximately 0.40 monolayer, in good agreement with charge density results. The cyclic voltammetry experiments demonstrated that the amount of bulk Se obtained during the potential scan to approximately 2 Se monolayers, which was further confirmed by electrochemical quartz crystal microbalance (EQCM) measurements that pointed out a mass variation corresponding of 3 monolayers of Se. In addition, the Se thin films were obtained by chronoamperometric experiments, where the Au electrode was polarized at +0.10V during different times in 1.0 M H(2)SO(4) + 1.0 mM SeO(2). The topologic aspects of the electrodeposits were observed in Atomic Force Microscope (AFM) measurements. Finally, in highly negative potential polarizations, the H(2)Se formation was analyzed by voltammetric and nanogravimetric measurements. These finding brings a new light on the selenium electrodeposition and point up to a proposed electrochemical model for molecule controlled surface engineering. (c) 2009 Elsevier Ltd. All rights reserved.

Podcasting Oral Presentations

This presentation tells how podcasts can be used to enhance the learning experience of English as a Second Language (ESL) students registered in a content-based language immersion program. The students were placed in groups of four and asked to prepare an oral presentation. The topics of the presentations included Financial Aid, College Courses and Course Schedules and Academic Policies. Each presentation was podcasted to give students an opportunity for self-evaluation and feedback.

Abstracts of Presentations

Abstracts of the 2008 Colby Undergraduate Research Symposium Presentations.