Computer algebra systems, such as GAP and Magma, are widely used for study­ing groups. Practical algorithms underlying these systems have been under develop­ment for over 30 years. More recently, there have been deep theoretical investigations into the asymptotic complexity of group-theoretic problems. The results have in­cluded demonstrations of polynomial-time solvability of problems whose traditional implementation, though usually efficient, would require exponential time in the worst case. This dissertation focuses on deterministic algorithms that meet both practical and theoretical standards of efficiency.

Most permutation-group computation employs a point-stabilizer series, a data structure first suggested by Sims in the 1960s. Variations by Knuth and Jerrum of Sims's basic algorithm had been shown to run in worst-case time O(sn² + n⁵), where n is the size of the permutation domain and s is the number of generators for the group. Certain important questions about the permutation group G , however, can be answered substantially faster than the time needed for just setting up Sims's datastructure. We present such fast algorithms. Testing nilpotence of a group is shown to have deterministic time complexity O(sn log n log n). Solvability is shown to be testable in time O(sn²).

Data structures are developed for computations in the families of nilpotent and solvable permutation groups. While reflecting the normal series that characterize the respective group family, these data structures are also naturally constructed and viewed within the permutation domain. Furthermore, they can be computed faster than the point-stabilizer series. The effectiveness of the data structures is demon­strated in their facilitation of algorithms that are based on the use of normal series.

For subgroups G and H of a nilpotent group, we consider computation of the following subgroups: the normalizer of H in G; the intersection of H and G; the centralizer of H in G.

The use of the data structure for solvable groups is illustrated in the implemen­tation of a method for finding Sylow subgroups. It makes essential use of the vector space representation of the factors in the normal series.

All algorithms have been implemented in GAP and proved to perform well in practice, especially the recognition algorithms.