Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

%_ **************************************************************************
%_ * The TeX source for AMS journal articles is the publisher's TeX code    *
%_ * which may contain special commands defined for the AMS production      *
%_ * environment.  Therefore, it may not be possible to process these files *
%_ * through TeX without errors.  To display a typeset version of a journal *
%_ * article easily, we suggest that you either view the HTML version or    *
%_ * retrieve the article in DVI, PostScript, or PDF format.                *
%_ **************************************************************************
% Author Package file for use with AMS-LaTeX 1.2
\controldates{10-FEB-1998,10-FEB-1998,10-FEB-1998,10-FEB-1998}
 
\documentclass{era-l}
\issueinfo{4}{01}{}{1998}
\dateposted{February 13, 1998}
\pagespan{1}{3}
\PII{S 1079-6762(98)00039-0}
\def\copyrightyear{1998}

%\issueinfo{4}% volume number
%  {1}%         % issue number
%  {January}%         % month
%  {1998}%     % year
  
%\copyrightinfo{1998}{American Mathematical Society}  

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\numberwithin{equation}{section}

\begin{document}

\title{Some Lie rings associated with Burnside groups} 

% author one information
\author{M. F. Newman}
\address{Australian National University}
%\curraddr{}
\email{newman@maths.anu.edu.au}
%\thanks{}

% author two information
\author{Michael Vaughan-Lee}
\address{University of Oxford}
\email{vlee@maths.oxford.ac.uk}
%\thanks{}

%\subjclass{17-04,17B30,20D15}
\subjclass{Primary 17-04, 17B30, 20D15}
%\date{November 1997}
\date{November 20, 1997}

\commby{Efim Zelmanov}

%\dedicatory{}

% presently the "communicated by" line appears only in PROC
%\commby{}

\begin{abstract}
%
We describe some calculations in graded Lie rings which provide 
a fairly sharp upper bound for the nilpotency class and 
for the order of the restricted Burnside group 
on two generators with exponent 7.
%
\end{abstract}

\maketitle

The purpose of this announcement is to report on some, machine-based,
calcula\-tions in graded Lie rings which provide a fairly sharp upper bound 
for the nilpotency class and for the order of
the restricted Burnside group $R(2,7)$ on two generators with exponent 7.

The associated Lie ring of $R(2,7)$ may be viewed as a Lie algebra
over the field ${\mathbb Z}_7$ with 7 elements. 
Furthermore, it is known to satisfy certain multilinear identities, 
the most significant of which is equivalent to the 6-Engel identity.
Additional multilinear identities are described in \cite{vlee}
(Section 2.5).
We let $E(2,7)$ be the largest two-generator Lie algebra over 
${\mathbb Z}_7$ satisfying the 6-Engel identity, and 
let $W(2,7)$ be the largest two-generator Lie algebra over ${\mathbb Z}_7$ 
satisfying all the multilinear identities described in \cite{vlee}.
So the associated Lie ring of $R(2,7)$ is a homomorphic image of
$W(2,7)$, which is itself a homomorphic image of $E(2,7)$.

Earlier calculations, described in \cite{hnvl}, 
showed that the largest class 18 quotient of $E(2,7)$ has dimension 6473
and  the largest class 18 quotient of $W(2,7)$ has dimension 6366. 
This result required the use of a multilinear identity of degree 13
first pointed out by Wall \cite{wall}.
This, together with results of Wall \cite{wall2},
implies that the largest class 18 quotient of $R(2,7)$ has order $7^{6366}$.

At that time Havas {\it et al.} \cite{hnvl} 
made the guess that the class of $W(2,7)$ is 
at least 30 and its dimension is at least $20000$. 
(In \cite{vlee} a more conservative estimate is given: the order
of $R(2,7)$ is at least $7^{10000}$.)

A rather astronomical upper bound for the class of $R(m,7)$ was 
obtained by %\mbox{
Vaughan-Lee %}
 and Zelmanov \cite{vlz} (Theorem 4).
More recently Vaughan-Lee \cite{vlee1} has obtained 
the reasonably moderate upper bound $51m^8$ for the class of $R(m,7)$. 
Moreover, these methods, augmented by appropriate computations, 
can be pushed quite a bit further to give bounds 
in the hundreds for the class of $R(2,7)$.

With recent improvements in computer hardware and the program,
it seemed worth making further calculations in the spirit of
\cite{hnvl}.
For example, the program now allows closure under 
automorphisms---as described in \cite{mfnob} for the group case.
With some additional refinements---in particular, the use of a
more compact data structure suggested by George 
Havas---it has turned out to be possible to show that $E(2,7)$ has
class 29 and dimension 23789. 

Showing these are upper bounds turns out to require 
only moderate resources, 
%\newpage
%\noindent
at most 120 MB and about 30 hours on a Dec Alpha. 
To confirm that the product presentation obtained this way 
is consistent and that the algebra so defined satisfies 
the 6-Engel condition is, of course, much more demanding. 
It takes about 200 hours depending on the machine used
(in our case various machines). 

Moreover, we have taken the next step. 
First enforce instances of Wall's degree 13 multilinear identity 
in addition to the 6-Engel identity. 
This gives an upper bound of 20418 for the dimension of $W(2,7)$. 
The product presentation so defined has class 29 and 
can be stored in about 90 MB.
Confirming that this algebra satisfies Wall's multilinear identity
requires further lengthy computations.
It then only needs a few computations to show that it satisfies
all the multilinear identities of $R(2,7)$
and hence that $W(2,7)$ does indeed have dimension 20418 and class 29.

These calculations show that $R(2,7)$ has order at most $7^{20418}$. 
It seems likely that the actual order will be close to this bound.
 
We have begun the lengthy task of determining the order of $R(2,7)$.

The ranks of the homogeneous components from weight 18 are given 
in the table 
(the ranks for lower weights are given in \cite{vlee}, Section 7.3).

%\bigskip

%\begin{center}

\begin{table}[h]
\begin{tabular}{|r|r|r|r|r|}
\hline
& \multicolumn{2}{c|} {$ E(2,7)$} & \multicolumn{2}{c|} {$W(2,7)$} \\ \hline
Class & Rank & Total & Rank & Total \\ \hline
18 & 2221 & 6473 & 2157 & 6366 \\
19 & 3136 & 9609 & 2992 & 9358 \\
20 & 4104 & 13713 & 3795 & 13153 \\
21 & 4716 & 18429 & 4046 & 17199 \\
22 & 4039 & 22468 & 2850 & 20049\\
23 & 1192 & 23660 & 240 & 20289 \\
24 & 96 & 23756 & 96 & 20385 \\
25 & 14 & 23770 & 14 & 20399 \\
26 & 14 & 23784 & 14 & 20413 \\
27 & 2 & 23786 &  2 & 20415 \\
28 & 1 & 23787 & 1 & 20416 \\
29 & 2 & 23789 & 2 & 20418 \\ \hline
\end{tabular}
\end{table}
%\end{center}

%\bigskip

Let $E[i,j]$ be the multiweight component of $E(2,7)$ spanned by
the Lie products of weight $i$ in the first generator and weight
$j$ in the second generator. The calculations show that the
multiweight components $E[i,j]$ with $i \ge 16$ are trivial.
This suggests that perhaps the normal closure of an element 
in $R(m,7)$ has class 15 and so the class of $R(m,7)$ is at most $15m$. 
Calculations to settle this seem to be out of reach at present.

\section*{Acknowledgements} 

This work was partly supported by grants for visits
from the Mathematical Sciences Research Visitor Program of the 
Australian National University and the Engineering and Physical
Sciences Research Council (EPSRC) of the United Kingdom 
(grant number GR/L36079),
and by grants for computing from EPSRC (grant number GR/L60326) and 
the ANU Supercomputing Facility.

%\medskip
%\newpage

\begin{thebibliography}{99}

\bibitem{hnvl}  George Havas, M. F. Newman, and M. R. Vaughan-Lee, 
{\em A nilpotent quotient algorithm for graded Lie rings}, 
J. Symbolic Comput. {\textbf 9} (1990), 653--664.
\MR{92d:20054}
\bibitem{mfnob}  M. F. Newman and E. A. O'Brien, 
{\em Application of computers to questions like those of Burnside.} II, 
Internat. J. Algebra Comput. {\textbf 6} (1996), 593--605.
\MR{97k:20002}
\bibitem{vlee}  Michael Vaughan-Lee, {\em The restricted Burnside problem},
London Mathematical Society Monographs, New Series, vol. 8, 2nd ed.,
Clarendon Press, Oxford, 1993.
\MR{98b:20047}
\bibitem{vlee1} Michael Vaughan-Lee, 
{\em The nilpotency class of finite groups of exponent $p$}, 
Trans. Amer. Math. Soc. \textbf{346} (1994), 617--640.
\MR{95g:20021}
\bibitem{vlz} Michael Vaughan-Lee and E. I. Zelmanov,
{\em Upper bounds in the restricted Burnside problem},
J. Algebra \textbf{162} (1993), 107--145.
\MR{94j:20019} 
\bibitem{wall}  G. E. Wall, {\em On the Lie ring of a group of prime exponent},
Proc. Second Internat. Conf. Theory of Groups, Canberra, 1973, pp. 667--690,
Lecture Notes in Math., vol. 372, Springer-Verlag, Berlin, Heidelberg,
New York, 1974.
\MR{50:10098}
\bibitem{wall2}  G. E. Wall, {\em On the Lie ring of a group of prime 
exponent.} II,
Bull. Austral. Math. Soc. \textbf{19} (1978), 11--28.              
\MR{80b:20052}
\end{thebibliography}

\end{document}



\endinput
30-Jan-98 08:05:09-EST,288147;000000000000
Return-path: 
Received: from AXP14.AMS.ORG by AXP14.AMS.ORG (PMDF V5.1-8 #1)
 id <01ISZBBY1KKW001B6M@AXP14.AMS.ORG>; Fri, 30 Jan 1998 08:05:06 EST
Date: Fri, 30 Jan 1998 08:05:05 -0500 (EST)
From: "pub-submit@ams.org " 
Subject: ERA/9800
To: pub-jour@MATH.AMS.ORG
Reply-to: pub-submit@MATH.AMS.ORG
Message-id: <01ISZBJDEUJK001B6M@AXP14.AMS.ORG>
MIME-version: 1.0
Content-type: TEXT/PLAIN; CHARSET=US-ASCII


Return-path: 
Received: from gate1.ams.org by AXP14.AMS.ORG (PMDF V5.1-8 #1)
 with SMTP id <01ISYCSNBVWG0014TQ@AXP14.AMS.ORG>; Thu, 29 Jan 1998 15:30:29 EST
Received: from leibniz.math.psu.edu ([146.186.130.2])
 by gate1.ams.org via smtpd (for axp14.ams.org [130.44.1.14]) with SMTP; Thu,
 29 Jan 1998 20:30:00 +0000 (UT)
Received: from gauss.math.psu.edu (aom@gauss.math.psu.edu [146.186.130.196])
 by math.psu.edu (8.8.5/8.7.3) with ESMTP id PAA28469 for
 ; Thu, 29 Jan 1998 15:29:58 -0500 (EST)
Received: (aom@localhost) by gauss.math.psu.edu (8.8.8/8.6.9)
 id PAA06772 for pub-submit@MATH.AMS.ORG; Thu, 29 Jan 1998 15:29:54 -0500 (EST)
Date: Thu, 29 Jan 1998 15:29:54 -0500 (EST)
From: Alexander O Morgoulis 
Subject: *accepted to ERA-AMS, Volume 4, Number 1, 1997*
To: pub-submit@MATH.AMS.ORG
Message-id: <199801292029.PAA06772@gauss.math.psu.edu>

begin 664 1998-01-001.tar
M,3DY."TP,2TP,#$N9'9I````````````````````````````````````````
M````````````````````````````````````````````````````````````
%% Modified January 23, 1998 by A. Morgoulis

%------------------------------------------------------------------------------
% Beginning of journal.top
%------------------------------------------------------------------------------
% This is a journal topmatter template file for use with AMS-LaTeX.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

% Date: 30-JAN-1998
%   \pagebreak: 0   \newpage: 2   \displaybreak: 0
%   \eject: 0   \break: 0   \allowbreak: 0
%   \clearpage: 0   \allowdisplaybreaks: 0
%   $$: 0   {eqnarray}: 0
%   \linebreak: 0   \newline: 0
%   \mag: 0   \mbox: 1   \input: 0
%   \baselineskip: 0   \rm: 0   \bf: 1   \it: 1
%   \hsize: 0   \vsize: 0
%   \hoffset: 0   \voffset: 0
%   \parindent: 0   \parskip: 0
%   \vfil: 0   \vfill: 0   \vskip: 0
%   \smallskip: 0   \medskip: 1   \bigskip: 2
%   \sl: 0   \def: 0   \let: 0   \renewcommand: 0
%   \tolerance: 0   \pretolerance: 0
%   \font: 0   \noindent: 1
%   ASCII 13 (Control-M Carriage return): 0
%   ASCII 10 (Control-J Linefeed): 0
%   ASCII 12 (Control-L Formfeed): 0
%   ASCII 0 (Control-@): 0
%