Computation with Finitely Presented Groups

This book describes the basic algorithmic ideas behind current methods for computing with finitely presented groups.

Computation with Finitely Presented Groups

Research in computational group theory, an active subfield of computational algebra, has emphasised three areas: finite permutation groups, finite solvable groups, and finitely presented groups. This book deals with the third of these areas. The author emphasises the connections with fundamental algorithms from theoretical computer science, particularly the theory of automata and formal languages, computational number theory, and computational commutative algebra. The LLL lattice reduction algorithm and various algorithms for Hermite and Smith normal forms from computational number theory are used to study the abelian quotients of a finitely presented group. The work of Baumslag, Cannonito and Miller on computing nonabelian polycyclic quotients is described as a generalisation of Buchberger's Gröbner basis methods to right ideals in the integral group ring of a polycyclic group. Researchers in computational group theory, mathematicians interested in finitely presented groups and theoretical computer scientists will find this book useful.

Methods for Investigating Finitely presented Groups

This thesis concerns computation with finite group presentations and its application to resolve certain open problems of Kim and Kostrikin.

Methods for Investigating Finitely presented Groups

This thesis concerns computation with finite group presentations and its application to resolve certain open problems of Kim and Kostrikin. We use the low-index subgroups and abelian quotient invariants algorithms to show that each member of a certain family of finitely-pesented groups is infinite. We present a new algorithm for determining which generalised dihedral groups are quotients of a given finitely-presented group, and use this to show that the groups in another family are pairwise non-isomorphic. Also we describe the method of 'pictures' over group presentations, discussing what they represent and how they can be used to obtain information about the group.

Computational Support for Discrete Mathematics

and Theoretical Computer Science Volume 15 , 1994 Application of Computational Tools for Finitely Presented Groups GEORGE HAVAS AND EDMUND F. ROBERTSON ABSTRACT . Computer based techniques for recognizing finitely presented groups are ...

Computational Support for Discrete Mathematics

With recent technological advances in workstations, graphics, graphical user interfaces, and object oriented programming languages, a significant number of researchers are developing general-purpose software and integrated software systems for domains in discrete mathematics, including graph theory, combinatorics, combinatorial optimization, and sets. This software aims to provide effective computational tools for research, applications prototyping, and teaching. In March 1992, DIMACS sponsored a workshop on Computational Support for Discrete Mathematics in order to facilitate interactions between the researchers, developers, and educators who work in these areas. Containing refereed papers based on talks presented at the workshop, this volume documents current and past research in these areas and should provide impetus for new interactions.

Handbook of Computational Group Theory

The methods to be discussed in the final two chapters of this book provide a completely different approach to computing with finitely presented groups. Their principal aim is to find a normal form for group elements, together with an ...

Handbook of Computational Group Theory

The origins of computation group theory (CGT) date back to the late 19th and early 20th centuries. Since then, the field has flourished, particularly during the past 30 to 40 years, and today it remains a lively and active branch of mathematics. The Handbook of Computational Group Theory offers the first complete treatment of all the fundame

Elementary Theory of Groups and Group Rings and Related Topics

... T. Daly, J. Niu); Use finitely presented groups as a replacement for social security numbers, for access control in data bases and electronic communications, leveraging computations carried out on hand-held devices running MAGNUS ...

Elementary Theory of Groups and Group Rings  and Related Topics

This proceedings volume documents the contributions presented at the conference held at Fairfield University and at the Graduate Center, CUNY in 2018 celebrating the New York Group Theory Seminar, in memoriam Gilbert Baumslag, and to honor Benjamin Fine and Anthony Gaglione. It includes several expert contributions by leading figures in the group theory community and provides a valuable source of information on recent research developments.

Computational and Geometric Aspects of Modern Algebra

Application of computational tools for finitely presented groups . In Computational support for discrete mathematics , DIMACS Ser . Discrete Math . Theoret . Comput . Sci . 15 ( 1994 ) 29–39 . [ 9 ] Derek F. Holt and Sarah Rees .

Computational and Geometric Aspects of Modern Algebra

A collection of papers from leading researchers in algebra and geometric group theory.

Groups and Computation II

The solution of the conjugacy problem for hyperbolic groups is not as easy to describe . It suffices here to say that it does provide a model for experimentation for finitely presented groups as a whole . There is no algorithm whereby ...

Groups and Computation II


Algorithmic Algebra and Number Theory

D. F. Holt, W. Plesken, A cohomological criterion for a finitely presented group to be infinite, J. London Math. Soc. ... S. Rees, A graphical system for displaying finite quotients offinitely presented groups, Groups and Computations, ...

Algorithmic Algebra and Number Theory

This book contains 22 lectures presented at the final conference of the Ger man research program (Schwerpunktprogramm) Algorithmic Number The ory and Algebra 1991-1997, sponsored by the Deutsche Forschungsgemein schaft. The purpose of this research program and of the meeting was to bring together developers of computer algebra software and researchers using com putational methods to gain insight into experimental problems and theoret ical questions in algebra and number theory. The book gives an overview on algorithmic methods and on results ob tained during this period. This includes survey articles on the main research projects within the program: • algorithmic number theory emphasizing class field theory, constructive Galois theory, computational aspects of modular forms and of Drinfeld modules • computational algebraic geometry including real quantifier elimination and real algebraic geometry, and invariant theory of finite groups • computational aspects of presentations and representations of groups, especially finite groups of Lie type and their Heeke algebras, and of the isomorphism problem in group theory. Some of the articles illustrate the current state of computer algebra sys tems and program packages developed with support by the research pro gram, such as KANT and LiDIA for algebraic number theory, SINGULAR, RED LOG and INVAR for commutative algebra and invariant theory respec tively, and GAP, SYSYPHOS and CHEVIE for group theory and representation theory.

Computational Algebra and Number Theory

Computational Group Theory, (Durham, 1982), London: Academic Press, 1984, pp. 145–183. ... [7] S. A. Linton, Constructing Matrix Representations of Finitely Presented Groups, J. Symbolic Comput. 124&5 (1991), 427-438.

Computational Algebra and Number Theory

Computers have stretched the limits of what is possible in mathematics. More: they have given rise to new fields of mathematical study; the analysis of new and traditional algorithms, the creation of new paradigms for implementing computational methods, the viewing of old techniques from a concrete algorithmic vantage point, to name but a few. Computational Algebra and Number Theory lies at the lively intersection of computer science and mathematics. It highlights the surprising width and depth of the field through examples drawn from current activity, ranging from category theory, graph theory and combinatorics, to more classical computational areas, such as group theory and number theory. Many of the papers in the book provide a survey of their topic, as well as a description of present research. Throughout the variety of mathematical and computational fields represented, the emphasis is placed on the common principles and the methods employed. Audience: Students, experts, and those performing current research in any of the topics mentioned above.

Groups St Andrews 2005

Indeed, many questions about finitely presented groups are un- solvable in general. Algorithms exist for answering ... Comprehensive details about computing with finitely presented groups appear in [32, 17]. Of particular relevance to ...

Groups St Andrews 2005

Selected papers from 'Groups St Andrews 2005' cover a wide spectrum of modern group theory.

Artificial Intelligence and Symbolic Mathematical Computation

PATCH Graphs : An Efficient Data Structure for Completion of Finitely Presented Groups Christopher Lynch and Polina Strogova ? 1 INRIA Lorraine - CRIN 2 INRIA Lorraine - CRIN and INRIA Rocquencourt Technopôle de Nancy - Brabois ...

Artificial Intelligence and Symbolic Mathematical Computation

Spine title: AISMC-3 : artificial intelligence and symbolic mathematical computation.

Geometric and Computational Perspectives on Infinite Groups

Your purchase supports the AMS' mission, programs, and services for the mathematical community. https://doi.org/10.1090/dimacs/025/09 Computing nilpotent quotients of finitely presented groups.

Geometric and Computational Perspectives on Infinite Groups

This book contains the proceedings of two workshops on computational aspects of geometric group theory. The workshops, held in the winter of 1994 at DIMACS and at the Geometry Center, covered practical group theoretic computation and theoretical problems. Containing both research and expository articles, this book is the only one available concentrating on the computational aspects of geometric group theory. Because this area involves an interplay between group theory, geometry, and automata theory, the expository articles in this book should help researchers in these fields to make connections to the other areas.

Groups St Andrews 1981

[22] Derek F. Holt, Bettina Eick and Eamonn A. O'Brien, Handbook of Computational Group Theory, CRC Press (2005). [23] Derek F. Holt and Sarah Rees, A graphics system for displaying finite quotients of finitely presented groups, ...

Groups   St Andrews 1981

This book contains selected papers from the international conference 'Groups - St Andrews 1981', which was held at the University of St Andrews in July/August 1981. Its contents reflect the main topics of the conference: combinatorial group theory; infinite groups; general groups, finite or infinite; computational group theory. Four courses, each providing a five-lecture survey, given by J. Neubuser (Aachen), D. J. S. Robinson (Illinois), S. J. Tobin (Galway) and J. Wiengold (Cardiff), have been expanded into articles, forming the first part of the book. The second part consists of surveys and research articles written by other conference participants. More than two-thirds of the book is composed of survey articles providing a remarkably clear and up-to-date picture of those areas of group theory. The articles which comprise this book, together with their extensive bibliographies, will prove an invaluable tool to researchers in group theory, and, in addition, their detailed expositions make them very suitable for relevant postgraduate courses.

Mathematical Software ICMS 2010

Computing Polycyclic Quotients of Finitely (L-)Presented Groups via Groebner Bases Bettina Eick and Max Horn Institut Computational Mathematics, TU Braunschweig, Pockelsstrasse 14, 38106 Braunschweig, Germany {beick ...

Mathematical Software   ICMS 2010

The ICMS Developer's Meeting is an international congress for which the main theme is mathematical software. The 2010 meeting was the third of a series of meetings of similar theme, the ?rst being held in Beijing, China in 2002,and the second in Castro-Urdiales, Spain in 2006. The ?eld of mathematics has numerous branches, and in each branch we ?nd that algorithms, and also implementations and applications of software s- tems, are studied. Researchers who endeavor to make such studies also have international meetings within their speci'c branches of mathematics, and these meetings have made signi'cant contributions to the ?elds in which they lie. The ICMS (International Congresseson Mathematical Software), on the other hand, is a general (not branch speci'c) meeting on mathematical software, which is held every four years, and is a rare opportunity for developers of mathematical softwarefrom di'erent branchesof mathematics, as well as mathematicians who are interested in mathematical software, to gather together.

Computational Group Theory and the Theory of Groups

AMS Special Session on Computational Group Theory, March 3-4, 2007, Davidson College, Davidson, North Carolina American Mathematical Society. ... To find the largest 2-quotient of the finitely presented group H of class ...

Computational Group Theory and the Theory of Groups

The power of general purpose computational algebra systems running on personal computers has increased rapidly in recent years. For mathematicians doing research in group theory, this means a growing set of sophisticated computational tools are now available for their use in developing new theoretical results. This volume consists of contributions by researchers invited to the AMS Special Session on Computational Group Theory held in March 2007. The main focus of the session was on the application of Computational Group Theory (CGT) to a wide range of theoretical aspects of group theory.The articles in this volume provide a variety of examples of how these computer systems helped to solve interesting theoretical problems within the discipline, such as constructions of finite simple groups, classification of $p$-groups via coclass, representation theory and constructions involving free nilpotent groups. The volume also includes an article by R. F. Morse highlighting applications of CGT in group theory and two survey articles. Graduate students and researchers interested in various aspects of group theory will find many examples of Computational Group Theory helping research and will recognize it as yet another tool at their disposal.

Finite Geometries Groups and Computation

Introduction Research activity in computational group theory has concentrated on four primary areas: permutation groups, finitely-presented groups, polycyclic groups, and representation theory. It is now possible in practice to study ...

Finite Geometries  Groups  and Computation

This volume is the proceedings of a conference on Finite Geometries, Groups, and Computation that took place on September 4-9, 2004, at Pingree Park, Colorado (a campus of Colorado State University). Not accidentally, the conference coincided with the 60th birthday of William Kantor, and the topics relate to his major research areas. Participants were encouraged to explore the deeper interplay between these fields. The survey papers by Kantor, O'Brien, and Penttila should serve to introduce both students and the broader mathematical community to these important topics and some of their connections while the volume as a whole gives an overview of current developments in these fields.

Computation with Linear Algebraic Groups

Computation with finitely presented groups. Cambridge: Cambridge University Press; 1994. PeterSlodowy. Die Theorie der optimalen Einparameteruntergruppen für instabile Vektoren. In Algebraische Transformationsgruppen und ...

Computation with Linear Algebraic Groups

Designed as a self-contained account of a number of key algorithmic problems and their solutions for linear algebraic groups, this book combines in one single text both an introduction to the basic theory of linear algebraic groups and a substantial collection of useful algorithms. Computation with Linear Algebraic Groups offers an invaluable guide to graduate students and researchers working in algebraic groups, computational algebraic geometry, and computational group theory, as well as those looking for a concise introduction to the theory of linear algebraic groups.

Groups and Computation

Workshop on Groups and Computation, October 7-10, 1991 Larry Finkelstein, William M. Kantor. and Theoretical Computer Science Volume 11 , 1993 A Graphics System for Displaying Finite Quotients of Finitely Presented Groups DEREK F. HOLT ...

Groups and Computation

This volume contains papers presented at the Workshop on Groups and Computation, held in October, 1991. The workshop explored interactions among four areas: symbolic algebra and computer algebra, theoretical computer science, group theory, and applications of group computation. The relationships between implementation and complexity form a recurrent theme, though the papers also discuss such topics as parallel algorithms for groups, computation in associative algebras, asymptotic behavior of permutation groups, the study of finite groups using infinite reflection groups, combinatorial searching, computing with representations, and Cayley graphs as models for interconnection networks.

Computational Group Theory and the Theory of Groups II

Following Sims's method for computationally showing a finitely presented group is nilpotent, we computationally complete the k = 5 and rank 3 case. While the results in this paper are not new and the computations discussed here almost ...

Computational Group Theory and the Theory of Groups  II

This volume consists of contributions by researchers who were invited to the Harlaxton Conference on Computational Group Theory and Cohomology, held in August of 2008, and to the AMS Special Session on Computational Group Theory, held in October 2008. This volume showcases examples of how Computational Group Theory can be applied to a wide range of theoretical aspects of group theory. Among the problems studied in this book are classification of p-groups, covers of Lie groups, resolutions of Bieberbach groups, and the study of the lower central series of free groups. This volume also includes expository articles on the probabilistic zeta function of a group and on enumerating subgroups of symmetric groups. Researchers and graduate students working in all areas of Group Theory will find many examples of how Computational Group Theory helps at various stages of the research process, from developing conjectures through the verification stage. These examples will suggest to the mathematician ways to incorporate Computational Group Theory into their own research endeavors.