A Framework for Parallel Disk-Based Computation for Very Large Groups

For the first time in computational group theory, parallel disk-based search algorithms are used. These algorithms emphasize streaming access while avoiding the high latency of disk. Search and enumeration is a recurring theme throughout computational group theory. Examples include orbit computation, permutation group membership (permutation group order, random element of a permutation group,etc.), condensation of matrix representations, generation of permutation representations from matrix representations, group intersection, centralizer, and normalizer. Without the use of disk, larger computations of this type would be infeasible. An analysis of these algorithms is presented. Formulas with application-specific parameters are derived that successfully predict each of their run times. Armed with these algorithms, we compute a permutation representation of the Baby Monster sporadic simple group using 8 terabytes of distributed disk. In addition, we compute a matrix condensation of Fischer’s group Fi23 using 2 terabytes of distributed disk. Finally, we present and implement a library with a well-developed API that allows computational group theory programmers to easily use the search algorithms presented in this paper.