This research is concerned with the structure and representations of algebraic groups and associated finite groups. The principal investigator will establish results on the structure and embedding of reductive subgroups of simple algebraic groups; work on linking the subgroup structure of finite groups to that of algebraic groups; study the representation theory of the symmetric groups over fields of positive characteristic; and study representations of finite groups of Lie type over a field of the defining characteristic. A group is an algebraic structure with a single operation. It appears in many areas of mathematics, as well as, physics and chemistry. The fundamental building blocks of finite groups are finite simple groups. One of the major results in mathematics of the past decade is the classification of the finite simple groups, the proof of which would require 10,000 to 15,000 journal pages. This research is aimed at using this classification in the study of arbitrary finite groups.
