Research in Group Theory

California Institute Of Technology, Pasadena CA

Investigators

Abstract

The mathematical concept of a group, a set together with an operation for combining two elements to produce another element, plays a central role in mathematics and its applications. The classification of the finite simple groups is one of the premier achievements of modern mathematics, providing the algebraic foundation for the study of symmetry of finite objects. The proof of the classification theorem, however, is very long and complicated, consisting of as many as ten thousand pages spread among hundreds of articles. There are at least two books, aimed at a lay audience, describing the human aspects of this effort. An emerging area of mathematics, studying objects called fusion systems, offers hope of simplifying the proof of the classification using an innovative approach that the investigator aims to implement. The research focuses on two projects. First, the work aims to extend the current description of the subgroup structure of the finite simple groups, and to use that information to establish a negative answer to an important question in universal algebra, open for over 35 years: Is each finite lattice an interval in the lattice of subgroups of some finite group? The second project aims to classify a large subclass of the class of simple saturated 2-fusion systems of component type. One consequence of the second project should be a simplification of the proof of the theorem classifying the finite simple groups.