9401399 Thompson This award supports research on finite groups, discrete groups and Galois groups. The principal investigator will determine if the trace spectrum of a uniformizing group of a 4-punctured sphere is determined by the punctures. Given a natural number n, he will see if SL(2,2^n) is the Galois group of a regular extension of Q(T), where Q denotes the rationals, such that at most 4 points of the T-line ramify. He will also consider for a given number field K if there is a finite set S(K) of finite simple groups with the property that every finite group G, all of whose irrecudible characters have values in K, has all its composition factors alternating groups or in S(K). A group is an algebraic structure with a single operation. It appears in many areas of mathematics, as well as physics and chemistry. One of the major problems in algebra is concerned with identifying those groups which can be represented as a Galois group over the rationals. This involves a blend of algebra, number theory and algebraic geometry.
