The group membership problem for permutation groups is perhaps the most important problem of computational group theory. Solution of this problem seems to depend intrinsically on constructing a strong generating set. Until now, recognizing if a set of generators is strong has been thought to be as hard as constructing a strong generating set from an arbitrary generating set. This paper shows ho...

We describe a method for searching for ideal symmetric solutions to the Prouhet-Tarry-Escott Problem. We report results of extensive searches for solutions of sizes up to 12. We found two solutions of size 10 that are smaller by two orders of magnitude than the solution found by A. Letac in the 1940s, which was the smallest size 10 solution known before our search. 1. The Prouhet-Tarry-Escott P...

One of the major research directions in computational group theory over the past 15 years has been the development of effective algorithms for the investigation of subgroups of GL(d, F ) where F is a finite field. We survey this work. 2000 Mathematics Subject Classification: 20C20, 20C40

Part I of this thesis presents methods for finding the primitive permutation groups of degree d, where 2500 ≤ d < 4096, using the O’Nan–Scott Theorem and Aschbacher’s theorem. Tables of the groups G are given for each O’Nan– Scott class. For the non-affine groups, additional information is given: the degree d of G, the shape of a stabiliser in G of the primitive action, the shape of the normali...