scholarly journals Computer aided determination of a Fibonacci group

1976 ◽  
Vol 15 (2) ◽  
pp. 297-305 ◽  
Author(s):  
George Havas
Keyword(s):  
Computer Aided ◽  
Group A ◽  
Coset Enumeration

The Fibonacci group F(2, 7) has been known to be cyclic of order 29 for about five years. This was first established by computer coset enumerations which exhibit only the result, without supporting proofs. The working in a coset enumeration actually contains proofs of many relations that hold in the group. A hand proof that F(2, 7) is cyclic of order 29, based on the working in computer coset enumerations, is presented here.

scholarly journals Some challenging group presentations

1999 ◽  
Vol 67 (2) ◽  
pp. 206-213 ◽  
Author(s):  
George Havas ◽  
Derek F. Holt ◽  
P. E. Kenne ◽  
Sarah Rees
Keyword(s):  
Finite Groups ◽  
Infinite Groups ◽  
Derived Length ◽  
Finitely Presented Group ◽  
Group Presentations ◽  
Coset Enumeration ◽  
Finitely Presented

AbstractWe study some challenging presentations which arise as groups of deficiency zero. In four cases we settle finiteness: we show that two presentations are for finite groups while two are for infinite groups. Thus we answer three explicit questions in the literature and we provide the first published deficiency zero presentation for a group with derived length seven. The tools we use are coset enumeration and Knuth-Bebdix rewriting, which are well-established as methods for proving finiteness or otherwise of a finitely presented group. We briefly comment on their capabilities and compare their performance.

scholarly journals On semigroup presentations

1993 ◽  
Vol 36 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Edmund F. Robertson ◽  
Yusuf Ünlü
Keyword(s):  
Group Presentation ◽  
Enumeration Algorithm ◽  
Long Period ◽  
Coset Enumeration ◽  
Finitely Presented

Semigroup presentations have been studied over a long period, usually as a means of providing examples of semigroups. In 1967 B. H. Neumann introduced an enumeration method for finitely presented semigroups analogous to the Todd–Coxeter coset enumeration process for groups. A proof of Neumann's enumeration method was given by Jura in 1978.In Section 3 of this paper we describe a machine implementation of a semigroup enumeration algorithm based on that of Neumann. In Section 2 we examine certain semigroup presentations, motivated by the fact that the corresponding group presentation has yielded interesting groups. The theorems, although proved algebraically, were suggested by the semigroup enumeration program.

scholarly journals Double-coset enumeration algorithm for symmetrically generated groups

2005 ◽  
Vol 2005 (5) ◽  
pp. 699-715 ◽  
Author(s):  
Mohamed Sayed
Keyword(s):  
Double Coset ◽  
Computer Implementation ◽  
Enumeration Algorithm ◽  
Coset Enumeration

A double-coset enumeration algorithm for groups generated by symmetric sets of involutions together with its computer implementation is described.

THE MANIFESTATION OF GROUP ENDS IN THE TODD–COXETER COSET ENUMERATION PROCEDURE

2007 ◽  
Vol 17 (01) ◽  
pp. 203-220
Author(s):  
ADAM PIGGOTT
Keyword(s):  
Sufficient Conditions ◽  
Coset Enumeration

The issue of recognizing group properties, such as the cardinality of the group, directly from the dynamics of an incomplete coset enumeration is discussed. In particular, it is shown that the property of having two ends is recognizable in such a way. Further, sufficient conditions are given for termination of a coset enumeration with the declaration that the group under consideration has infinitely-many ends.

scholarly journals Nice Efficient Presentions for all Small Simple Groups and their Covers

2004 ◽  
Vol 7 ◽  
pp. 266-283 ◽  
Author(s):  
Colin M. Campbell ◽  
George Havas ◽  
Colin Ramsay ◽  
Edmund F. Robertson
Keyword(s):  
Group Theory ◽  
Power Relations ◽  
Simple Groups ◽  
Computational Group Theory ◽  
The Status ◽  
Coset Enumeration ◽  
Covering Groups ◽  
Computational Properties ◽  
Selection Of ◽  
Better Than

AbstractPrior to this paper, all small simple groups were known to be efficient, but the status of four of their covering groups was unknown. Nice, efficient presentations are provided in this paper for all of these groups, resolving the previously unknown cases. The authors‘presentations are better than those that were previously available, in terms of both length and computational properties. In many cases, these presentations have minimal possible length. The results presented here are based on major amounts of computation. Substantial use is made of systems for computational group theory and, in partic-ular, of computer implementations of coset enumeration. To assist in reducing the number of relators, theorems are provided to enable the amalgamation of power relations in certain presentations. The paper concludes with a selection of unsolved problems about efficient presentations for simple groups and their covers.

scholarly journals Coset Enumeration Using Prefix Gröbner Bases: An Experimental Approach

2001 ◽  
Vol 4 ◽  
pp. 74-134
Author(s):  
Birgit Reinert ◽  
Dirk Zeckzer
Keyword(s):  
Experimental Approach ◽  
Gröbner Bases ◽  
New Method ◽  
Finitely Presented Groups ◽  
Monoid Ring ◽  
Rewriting System ◽  
Coset Enumeration ◽  
Basis Computation ◽  
String Rewriting ◽  
Finitely Presented

AbstractThe authors study a new method for coset enumeration in finitely presented groups. Their method uses prefix Gröbner basis computation in the monoid ring ${\mathbb{K}}[{\cal M}]$, where ${\mathbb{K}}$ is a computable field and ${\cal M}$ a monoid presented by a convergent string-rewriting system. The method is compared to well-known methods for Todd-Coxeter enumeration, using examples from the literature where studies of these methods are reported. New insights into coset enumeration were gained using three different kinds of orderings, combined with new frameworks and strategies implemented in MRC 1.2.

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