Todd–Coxeter algorithm

Todd–Coxeter algorithm
Name	Todd–Coxeter algorithm
Type	Algorithm
Field	Group theory
Introduced	1936
Developers	John A. Todd; H. S. M. Coxeter

Todd–Coxeter algorithm The Todd–Coxeter algorithm is a procedure for enumerating cosets of a subgroup in a finitely presented group. It provides a computational method to construct a coset table from generators and relators, enabling concrete calculations in combinatorial group theory and connections with algorithmic problems in algebra. The algorithm is fundamental in computational treatments of finite groups and interacts with many mathematical institutions and researchers in algebra and topology.

Introduction

The Todd–Coxeter algorithm was developed to enumerate left cosets of a subgroup H in a group G given by a presentation ⟨S | R⟩. It transforms symbolic presentations into explicit permutation representations and finite tables, facilitating work in computational algebra systems and collaborations among mathematicians at universities and laboratories. Principal users include researchers at institutions such as University of Cambridge, Princeton University, University of Chicago, Massachusetts Institute of Technology, and University of Oxford.

Historical background

John A. Todd and H. S. M. Coxeter published the algorithm amid active research in combinatorial group theory and Coxeter groups, following developments by contemporaries at University of Manchester and exchanges with scholars from Imperial College London. The method built on earlier work in coset enumeration and influenced subsequent projects at Bell Labs, IBM, and academic departments including Harvard University and Yale University. It inspired computational group theory initiatives at the National Physical Laboratory and shaped software efforts later realized at institutions such as University of Sydney and University of Waterloo.

Algorithm description

Given a presentation ⟨S | R⟩ and subgroup generators, the algorithm initializes a coset table and applies relators to deduce identifications and new cosets. Core operations include scanning relators, defining images under generators, performing coincidences, and closing the table by backtracking when contradictions arise. Implementations often appear in systems developed at University of Illinois Urbana–Champaign, University of California, Berkeley, California Institute of Technology, Cornell University, and ETH Zurich. Major adopters and contributors include researchers affiliated with Royal Society, American Mathematical Society, European Mathematical Society, Institute of Mathematics and its Applications, and national research councils.

Coset table and data structures

The coset table records rows indexed by cosets and columns indexed by generators; entries are images or undefined symbols to be filled. Efficient storage and update strategies borrow techniques from computer science groups at Stanford University, Carnegie Mellon University, University of Pennsylvania, University of Cambridge (Computer Laboratory), and research centers like Los Alamos National Laboratory. Handling coincidences uses union-find or disjoint-set data structures popularized in algorithmic contexts at Bell Labs and taught at Massachusetts Institute of Technology and Princeton University.

Examples and computations

Standard worked examples include enumerations for presentations related to classical groups and tessellation groups studied by H. S. M. Coxeter and others, with computations reproduced in texts from Cambridge University Press and Springer Science+Business Media. Concrete computations often illustrate the algorithm on presentations of finite symmetric groups, alternating groups, and triangle groups connected to researchers at University of Bonn, University of Göttingen, and Institut Henri Poincaré. Software packages implementing examples originated in projects at University of Sydney (GAP history contributors), Technische Universität Berlin, and collaborations with teams at Microsoft Research and Google Research.

Complexity and termination

Termination is guaranteed when the subgroup index is finite, yielding a finite coset table; otherwise the process may not terminate or may grow without bound. Complexity depends on the presentation, relator lengths, and subgroup generators; worst-case behavior can be exponential and is sensitive to choices that echo challenges faced in decision problems studied by researchers at Princeton University, University of California, San Diego, and Rutgers University. Analyses draw on work by scholars at University of Michigan, Columbia University, and the Institute for Advanced Study.

Variants and improvements

Many refinements address reduction of table size, strategy for scanning relators, and handling of coincidences. Notable variants include the HLT strategy, methods from coset enumeration literature developed at University of Cambridge, and heuristics studied by groups at University of Warwick and University of Leeds. Modern improvements integrate with algorithms from computational algebra packages originated by teams at University of St Andrews, Delft University of Technology, and University of Auckland.

Applications in group theory

The Todd–Coxeter algorithm is applied to compute permutation representations, subgroup indices, and to test properties of finitely presented groups in research undertaken at University of Oxford, University of Cambridge, Harvard University, Princeton University, and numerous mathematical departments worldwide. It aids in the classification of finite groups, the study of Coxeter groups and reflection groups associated with H. S. M. Coxeter and collaborators, and in constructing examples relevant to topology, geometry, and algebra studied at Institut de Mathématiques de Jussieu, Clay Mathematics Institute, Max Planck Institute for Mathematics, and other centers.

Category:Group theory