John J. Cannon

The MAGMA algebra system I: the user language

Abstract In the first of two papers on MAGMA , a new system for computational algebra, we present the MAGMA language, outline the design principles and theoretical background, and indicate its scope and use. Particular attention is given to the constructors for structures, maps, and sets.

Implementation and analysis of the Todd-Coxeter algorithm

A recent form of the Todd-Coxeter algorithm, known as the lookahead algorithm, is described. The time and space requirements for this algorithm are shown experimentally to be usually either equivalent or superior to the Felsch and Haselgrove- Leech-Trotter algorithms. Some findings from an experimental study of the behaviour of Todd-Coxeter programs in a variety of situations are given. 1. Introduction. The Todd-Coxeter algorithm (20) (TC algorithm) is a sys- tematic procedure for enumerating the cosets of a subgroup H of finite index in a group G, given a set of defining relations for G and words generating H. At the present time, Todd-Coxeter programs represent the most common application of computers to group theory. They are used for constructing sets of defining relations for particular groups, for determining the order of a group from its defining relations, for studying the structure of particular groups and for many other things. As an example of the use of the algorithm, consider the following family of defining relations, Men(n), due to Mennicke:

Construction of defining relators for finite groups

Given a faithful representation of a group G of order up to 10^4, we describe an algorithm, based on the notion of the graph of G, for constructing a concise presentation for G. This technique may be generalized to give a semialgorithm which is usually successful in finding presentations for groups of order up to 10^6.

Programming with algebraic structures: design of the MAGMA language

MAGMA is a new software system for computational algebra, number theory and geometry whose design is centred on the concept of algebraic structure (magma). The use of algebraic structure as a design paradigm provides a natural strong typing mechanism. Further, structures and their morphisms appear in the language as first class objects. Standard mathematical notions are used for the basic data types. The result is a powerful, clean language which deals with objects in a mathematically rigorous manner. The conceptual and implementation ideas behind MAGMA will be examined in this paper. This conceptual base differs significantly from those underlying other computer algebra systems.

Automorphism group computation and isomorphism testing in finite groups

A new method for computing the automorphism group of a finite permutation group and for testing two such groups for isomorphism is described. Some performance statistics are included for an implementation of these algorithms in the Magma language.

Computers in group theory: a survey

Computers are being applied to an increasingly diverse range of problems in group theory. The most important areas of application at present are coset enumeration, sub-group lattices, automorphism groups of finite groups, character tables, and commutator calculus. Group theory programs range from simple combinatorial or numerical programs to large symbol manipulation systems. In this survey the more important algorithms in use are described and contrasted, and results which have been obtained using existing programs are indicated. An extensive bibliography is included.

Computing maximal subgroups of finite groups

Abstract We describe a practical algorithm for computing representatives of the conjugacy classes of maximal subgroups in a finite group, together with details of its implementation for permutation groups in the MAGMA system. We also describe methods for computing complements of normal subgroups and minimal supplements of normal soluble subgroups of finite groups.

Computing in permutation and matrix groups. I. Normal closure, commutator subgroups, series

This paper is the first in a series which discusses computation in permutation and matrix groups of very large order. The fundamental concepts are defined, and some algorithms which perform elementary operations are presented. Algorithms to compute normal closures, commutator subgroups, derived series, lower central series, and upper central series are presented.

Computing the Subgroups of a Permutation Group

A new method for computing the conjugacy classes of subgroups of a finite group is described.

The Transitive Permutation Groups of Degree 32

We describe our successful computation of a list of representatives of the 2,801,324 conjugacy classes of transitive groups of degree 32.