Proving a group trivial made easy: A case study in coset enumeration

Abstract

Dedicated to B.H. Neumann, on the occasion of his 90th birthdayCoset enumeration, based on the methods described by Todd and Coxeter, is one of the basic tools for investigating finitely presented groups. The process is not well understood, and various pathological presentations of, for example, the trivial group have been suggested as challenge problems. Here we consider one such family of presentations proposed by B.H. Neumann. We show that the problems are much easier than they first appear, albeit at the expense of considerable preliminary 'experimentation'. This demonstrates how far the range of applicability of coset enumeration has improved.

Locally toroidal polytopes of rank 6 and sporadic groups

“…A number of comments on Theorem 1 are in order. The claim of finiteness of Γ depends upon coset enumeration, that was, for robustness purposes, carried out using two different implementations of the Todd-Coxeter algorithm: the built-in implementation of GAP system [10], and ACE implementation by G. Havas and C. Ramsay [13], also available as a GAP package [9]. Both computations in the case t = (3000), s = (2200) returned the index of the vertex stabiliser, the subgroup F := 3 4 :F 4 in the quotient Θ of the Coxeter group [3 2 , 4, 3 2 ] modulo the relations on Fig.…”

<title>Evolutionary algorithm in group theory</title>

“…Many of the problems were solved within 256 megabytes ofmain memory. This should be compared with the later experiments by Havas and Ramsay [12] [13] where an SGI Origin 2000 super computer was employed. That computer was equipped with more than 4 gigabytes of memory and could easily handle a coset table with 1 trillion table entries. Not only were results from the evolutionary algorithm on ACE achieved in a reasonable amount of time and space, it has been shown that the evolutionary algorithm described here can be used to verify existing hypothesizes.…”

On the Efficiency of Some Finite Groups

“…It is possible to enumerate billions of cosets [10], but such enumerations are computationally expensive. Hence, we use cheap filters to remove presentations which cannot define the desired group.…”