Research in Group Theory

California Institute Of Technology, Pasadena CA

Investigators

Abstract

Aschbacher will focus on two projects. First, he has begun a program to simplify parts of the classification of the finite simple groups, by working in the category of saturated 2-fusion systems. In particular Aschbacher hopes to obtain a new proof of the classification of simple groups of component type; this represents roughly half of the classification of the finite simple groups. A byproduct of this effort would be classifications of certain classes of fusion systems, perhaps including all simple saturated 2-fusion systems. Second, Aschbacher has begun a program to describe the subposet of the lattice of subgroups of a classical group consisting of the overgroups of certain important subgroups, such as quadratic subgroups and some tori. As an application, Aschbacher and J. Shareshian seek to prove there exist finite lattices which are not intervals in the subgroup lattice of any finite group. If successful, the work would lead to a new proof of one of the most important results in twentieth century mathematics, the classification of the finite simple groups, and to strong new results on the subgroup structure of the simple groups, the main tool for applying the classification. The simple groups are the building blocks of finite group theory, and finite groups are the mathematical tool with which one studies the symmetry of finite objects.