Efficient Computation in Finite Groups

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The proposed research is in the area of efficient manipulation of finite groups and estimation of their parameters. Potential application areas include computational group theory, graph isomorphism testing (of relevance to chemical documentation), efficient interconnection networks based on groups, and group-based cryptography.Our work belongs to the areas of the Theory of Computing, Group Theory, Symbolic Algebra, and Combinatorics. Building on our previous results in the complexity theory of group algorithms, we propose to pursue several directions of research.The main focus is the design and analysis of efficient algorithms for high degree per-mutation groups and for large dimensional matrix groups. We are looking for algorithms which satisfy both the requirements of fast asymptotic running time and good practical performance. In the permutation group setting, our nearly linear time algorithms achieved this goal for a quite broad class of algorithmic tasks; now we would like to extend this class of algorithms. We implemented most of our algorithms in the GAP programming language and they are available for the public as part of the standard library package of GAP. These algorithms represent the long-awaited marriage of theoretical and practical approaches to computational permutation group theory. We intend to continue the implementation effort.Our major goal is the first polynomial-time algorithm for the basic manipulation of arbitrary matrix groups. In matrix groups defined over a field of characteristic p, we would like to give a polynomial-time algorithm computing the order and a composition series, provided that we can compute discrete logarithms in the fields GF(pe ). Our recent algorithms for the constructive recognition of certain classes of finite simple groups are a major ingredient in this plan.Finally, we plan to investigate some "pure" algebraic and combinatorial problems, which are motivated by our algorithmic investigations or became more accessible through the methodological advances achieved in connection with our algorithmic results. In particular, we are interested in base size problems for permutation groups, problems concerning the action of groups on the power set of the permutation domain, and problems related to Cayley graphs: the diameter of Cayley graphs and the investigation of non-Cayley graphs with vertex-transitive automorphism group. Small bases are important for fast implementations and for improving the running time estimates of algorithms. Estimates of diameters of Cayley graphs are closely related to the expansion rate and through this to a host of basic questions of the Theory of Computing and Probability Theory.