GAP

GAP
Name	GAP
Latest release	4.12.3
Programming language	C (programming language), GAP language
Operating system	Linux, macOS, Microsoft Windows
Genre	Computer algebra system, Mathematical software
License	GNU General Public License

GAP is a computer algebra system for computational discrete algebra with particular emphasis on computational group theory, providing a programming language, libraries, and tools for researchers and educators. It supports computations with finite groups, permutation groups, matrix groups, algebras, and combinatorial structures, and integrates with a range of mathematical projects and databases. Developed and maintained by an international community, the system is widely used in research at institutions such as University of St Andrews, RWTH Aachen University, Technische Universität Braunschweig, and collaborations involving Mathematics of computation projects.

Overview

GAP offers a high-level interpreted language, an extensive library of functions, and a collection of contributed packages maintained by teams at University of Sydney, University of Glasgow, University of Kiel, University of Warwick, and other centers. The system includes databases of small groups, character tables, and transitive groups, used by researchers affiliated with American Mathematical Society meetings, European Mathematical Society conferences, and workshops at Institute for Advanced Study. Widely cited in papers in journals such as Journal of Algebra, Communications in Algebra, Experimental Mathematics, and Journal of Symbolic Computation, the software interacts with projects like SageMath, PARI/GP, Magma (software), and computational libraries developed at Max Planck Institute for Mathematics.

History

Origins of the system trace to collaborations between researchers at University of St Andrews and RWTH Aachen University in the late 1970s and 1980s, alongside efforts by teams at Universität Kiel and Technische Universität Dresden. Early releases incorporated algorithms developed in seminars at Institut des Hautes Études Scientifiques and implementations influenced by work from Berkeley, Cambridge, and Paris-Sud University groups. The project grew through contributions from researchers attending International Congress of Mathematicians satellite workshops, and milestones include integration of the Small Groups Library compiled by contributors linked to University of Oxford and additions stemming from collaborations with University of Sydney and Australian National University.

Structure and Classes

The system organizes mathematical objects into types and families implemented as records, groups, rings, modules, and other structures, with constructors and methods developed at institutions such as Imperial College London, University of California, Berkeley, and Princeton University. Core components provide representations for permutation groups, matrix groups, finitely presented groups, and algebraic objects used in research at University of Cambridge and Harvard University. The software’s package system allows extensions from contributors at University of Ljubljana, University of Zagreb, University of Bordeaux, and University of Antwerp, enabling specialized classes for homological algebra, representation theory, and combinatorial designs referenced in seminars at Institute for Mathematics and its Applications.

Algebraic Properties

Algorithms implemented include coset enumeration, Schreier–Sims, Todd–Coxeter, and computation of character tables, reflecting theoretical advances from researchers associated with École Normale Supérieure, University of Bonn, University of Chicago, University of Michigan, and Columbia University. The system facilitates computation of subgroup lattices, normalizer chains, and automorphism groups used in studies presented at American Mathematical Society sectional meetings and in doctoral research at University of Göttingen. Libraries include data compiled by contributors connected to University of Illinois at Urbana–Champaign and University of Toronto, enabling verification of conjectures in representation theory and group cohomology developed at Princeton and ETH Zurich.

Applications and Use Cases

Researchers employ the software in classification problems, constructing counterexamples, and verifying theorems in papers published in Annals of Mathematics, Proceedings of the London Mathematical Society, and Transactions of the American Mathematical Society. Use cases span computing Galois groups in algebraic number theory collaborations at University of Cambridge and University of Leiden, designing error-correcting codes in projects affiliated with Massachusetts Institute of Technology and École Polytechnique, and modelling symmetries in chemical graph theory studies linked to California Institute of Technology and University of Geneva. Educators at University College London, Yale University, and University of Illinois use the system for teaching courses on computational algebra and workshops at Mathematical Sciences Research Institute.

Implementations and Software Integration

The system’s core is implemented in C (programming language) and an interpreted language developed within the project, with interfaces and bindings created for SageMath, Python (programming language), PARI/GP, and Singular (computer algebra system). Packages are developed and distributed by teams at University of Stuttgart, University of Leeds, University of Birmingham, and University of Oslo, and integration efforts have been coordinated during coding sprints at European Research Council-funded centers and collaborations with National Science Foundation-supported projects. Cloud deployments and reproducible research workflows use containerization tools presented at USENIX and IEEE workshops, enabling deployment on infrastructure at Amazon Web Services and academic computing clusters at CERN.

Category:Computer algebra systems