The Computational Algebra Group has been in existence for nearly 25 years. During the 1970's Cannon and Butler developed many of the standard permutation group algorithms. Starting in the late 1970's, in joint work with Charles Leedham-Green, the Group commenced the design of algorithms for p-groups and soluble groups. In the 1980's interest shifted to representation theory, with algorithms for both ordinary and modular representations being developed by the Group in collaboration with Schneider (then at Essen) and others. In the 1990's, our interests have extended to number fields, elliptic curves, finite fields and computational linear algebra.
In the early 1970's, in conjunction with Neubüser's group in Aachen, the Computational Algebra Group developed the first publicly available software package for group theory, Group. In 1975, it commenced the development of the Cayley system for group theory and related areas. Following its release in 1982, Cayley was licensed to approximately 300 institutions. It has been successfully applied to a very wide variety of problems both within and outside mathematics.
A great deal of experience with algebraic structural computation was gained from Cayley in the context of computing structural information about groups. By 1985 it was clear that many of the lessons learnt here would carry over to other varieties of algebras. Initially, the plan was to produce a major upgrade of Cayley, but it soon became apparent that the new design was much too radical to be grafted onto Cayley. Consequently a completely new system (Magma) emerged. Distribution of Magma V1 commenced at the end of 1993 and by the beginning of 1995 it was running at approximately 200 sites in about 35 countries.