Mathematical research on computational group theory will be carried out by six students in this Research Experiences for Undergraduates project. Students will learn to apply the high level programming language CAYLEY in their investigations into group structure. This language is designed around algebraic concepts of structure, set and mapping. It contains an extensive data base of group theoretic and geometric information. Students define structures and then extract information about the structure. For example, a small finitely presented group (of order less than 20,000) can be completely analyzed by CAYLEY: conjugacy classes, subgroup lattice, normal subgroups and automorphism group. Work will focus on two areas, typical values and counting in groups. The former concept arises in probability when studying random samples. The sample means for subsamples may form groups with respect to symmetric differences. If the subsample means, or typical values, do form a group, then the intervals formed by the means contain the true mean with a known probability. The commutativity of group elements. This is a concept not usually accessible to researchers using CAYLEY. Questions such as counting the number of group elements which commute and calculating the probability that two elements commute will be taken up. Follow-up will provide encouragement and assistance to help students with publication and presentation of their results. Most will participate in the annual Rose-Hulman Undergraduate Mathematics Conference, begun in 1984.//
