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7 - Computing in groups with exponent six

Published online by Cambridge University Press:  06 July 2010

Michael Atkinson
Affiliation:
University of St Andrews, Scotland
Nick Gilbert
Affiliation:
Heriot-Watt University, Edinburgh
James Howie
Affiliation:
Heriot-Watt University, Edinburgh
Steve Linton
Affiliation:
University of St Andrews, Scotland
Edmund Robertson
Affiliation:
University of St Andrews, Scotland
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Summary

Abstract. We have investigated the nature of sixth power relations required to provide proofs of finiteness for some two-generator groups with exponent six. We have solved various questions about such groups using substantial computations. In this paper we elaborate on some of the calculations and address related problems for some three-generator groups with exponent six.

INTRODUCTION

Motivated by an aim to get estimates for the number and length of sixth power relations which suffice to define groups with exponent six, we studied finiteness proofs for presentations of such groups in. We tried to find relatively small sets of defining relations for various groups, with a view to improving our understanding of finiteness proofs.

We denote the free group on d generators with exponent n by B(d, n) and generally use notation as in. One question we would very much like to be able to answer is whether B(2,6) can be defined without using too many sixth powers. Here we focus on the computational components of the process, giving sample code which solves some associated problems.

We showed that B(2,6) has a presentation on 2 generators with 81 relations, which is derived from a polycyclic presentation. Here in Section 3 we give a program to construct a polycyclic presentation for B(2,6) which shows the structure of the group. If only sixth power relations are used, we showed that M. Hall's finiteness proof yields that fewer than 2124 sixth powers can define B(2,6). On the other hand the best lower bound we have proved is that at least 22 sixth powers are needed [5, Theorem 1].

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Publisher: Cambridge University Press
Print publication year: 2000

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