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Let M be a subgroup of a finite group G, and suppose that M normalizes the nilpotent residual \(H^\infty \) of every non-subnormal subgroup H of G. We show that M must also normalize the nilpotent residuals of the subnormal subgroups of G. We also prove a similar result for the solvable residual.     

Injective restriction of characters

Suppose that G is a finite group and let H be a proper subgroup of G. Then the restriction map from the characters of G to the characters of H is never injective.     

The Cancer Genetics Network: recruitment results and pilot studies

OBJECTIVE: The National Cancer Institute established the Cancer Genetics Network (CGN) to support collaborative investigations into the genetic basis of cancer susceptibility, explore mechanisms to integrate this new knowledge into medical practice, and identify ways of addressing the associated psychosocial, ethical, legal, and public health issues. SUBJECTS AND METHODS: The CGN has developed the complex infrastructure required to support the projects, including the establishment of guidelines and policies, uniform methods, standard questionnaires to be used by all of the centers, and a standard format for submission of data to the Informatics Center. Cancer patients and their family members have been invited to enroll and be included in a pool of potential study participants. The Information Technology Group is responsible for support of the design, implementation, and maintenance of the multicenter Network-wide research protocols. RESULTS: As of January 2004, the CGN contained data on 23,995 probands (participants) and 425,798 family members. As a resource for cancer genetic studies, the CGN has a large number of probands and first-degree relatives with and without cancer and with multiple ethnicities. Different study designs can be used including case-control, case-case, and family studies. CONCLUSIONS: The unique resources of the CGN are available for studies on cancer genetic susceptibility, translational research, and behavioral research. The CGN is now at a point where approved collaborators may have access to enrolled patients and their families for special studies, as well as to the clinical, environmental and family cancer history data banked in the Informatics Center.     

A life cycle framework for the investigation of environmentally benign nanoparticles and products

While significant advances in our understanding of the behavior of engineered nanoparticles in the environment continue, there remains a need to engage the nanoparticle research community directly in the development and evaluation of environmentally benign nanoparticles to ensure that nanomaterial‐based industries emerge as tools for sustainability rather than environmental liabilities. Current research efforts aimed at understanding the environmental implications of nanotechnology emphasize existing groups of nanoparticles and products already in commercial distribution. While this is clearly necessary, this approach fails to identify and address the many tradeoffs associated with product performance and environmental quality. We believe this to be a critical gap in the ongoing exploration of nanostructured materials and their properties and applications. We posit that a number of issues are not being holistically addressed, including resource availability and allocation, manufacturing energy requirements and embodied energy, material efficiency, environmental properties of nanomaterials and nanoproducts, and waste generation. An interdisciplinary approach to research, based on the life cycle paradigm and devoted to the identification, investigation, synthesis, testing, and analysis of groups of new, more environmentally conscious nanoparticles is needed. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nafion/tetraruthenated porphyrin glassy carbon-modified electrode: characterization and voltammetric studies of sulfite oxidation in water–ethanol solutions

In this work, the modification of a glassy carbon electrode with tetraruthenated porphyrins electrostatically assembled onto a Nafion film, previously adsorbed on the electrode surface, is reported. This modified electrode was characterized by scanning electron microscopy–energy-dispersive X-ray, Raman spectroscopy, UV-Vis spectroelectrochemistry, and cyclic voltammetry. The Nafion film onto the glassy carbon electrode shows a smooth disposition; when the tetraruthenated porphyrin is incorporated on the Nafion film, the complex is adsorbed in a homogeneous way. The modified electrode catalyzes HSO₃⁻ oxidation in water–ethanol solutions and shows an enhanced stability compared with the electrode modified with the dip coating method. Rotating disk electrode experiments showed a kinetic limitation to the electron transfer controlled by charge propagation in the film. I/E curves show a Tafel slope of 120 mV/decade corresponding to a first electron-transfer reaction, depending on the potential, as the determining step. Spectroelectrochemical experiments demonstrated that Ru(II) is the active site for the electrocatalysis.     

Solvable groups contain large centralizers

It is proved that every nonabelian solvable group contains a noncentral element whose centralizer has order exceeding its index. Research partially supported by a grant from the National Science Foundation.