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\title{\huge Proving another group infinite.}
\author{
Geoff C. Smith\\
School of Mathematical Sciences, University of Bath, Bath BA2 7AY, England
}
\date{}
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\maketitle

Let $G = F(4,7)$ denote the $7$ generator group with 4-step Fibonacci 
defining relations. To be explicit, $G$ has generators 
$\{g_i\ |\ i \in \Z/7\Z \}$ and relators $g_ig_{i+1}g_{i+2}g_{i+3}g_{i+4}^{-1}$
for each $i \in \Z/7\Z.$
\par THEOREM {\em The group $G$ is infinite.}
\par The cardinality of this group was not hitherto known. The groups
$F(r,n)$ have been intensively studied, and many families are reasonably
well understood. For example, $F(2,n)$ is finite only when $n$ is
$1,2,3,4,5$ or $7.$ For large $n$ this was proved by Lyndon (unpublished) 
by using small cancellation theory. For very small $n$ the results can
be obtained by hand using ingenuity or by the Todd-Coxeter coset
enumeration process. The latter technique is best left to a machine.

For moderate values of $n$ the issue was resolved by
Brunner for 8 and 10, and for $n=9$ by M.F. Newman \cite{Ne} using an ingenious 
embellishment of the Golod-\v{S}aferevi\v{c} theorem \cite{GN}.

Various families $F(r,n)$ with $r > 2$ are of known cardinality,
and something of the structure of some of the finite ones is understood 
\cite{CR}.  The literature is extensive, but
we draw the reader's attention the survey article of Thomas \cite{Th} which
gives an excellent description of the field.

The group studied in this paper does not fall into these understood families.
We will show that $G$ is infinite by appeal to a slightly strengthened
form of the Golod-\v{S}aferevi\v{c} theorem.
We state it now, in the version which appears in \cite{Jo}
(given the remarks in section 15.3 of that work).
\par THEOREM (Golod-\v{S}aferevi\v{c})
{\em Let $G = <X\ |\ R>$ be a finite group with $|X| = d$
and $|R| = r.$ Suppose that
$p$ is a rational prime with the property that $G/G'G^p$ has order $p^d,$
then $r > d^2/4.$}
\par The proof of our theorem depends on finding a subgroup of $G$ to which
we may apply the Golod-\v{S}aferevi\v{c} theorem. Such a subgroup was found
with the help of both CAYLEY \cite{Ca} and QUOTPIC \cite{HR}. 
The former is a well-known
Computational Group Theory system. The latter is a new group theory
graphics system created by Holt and Rees. 

Let $M = <a,b \ |\ a^3, b^7, b^a = b^2>$ be a non-abelian group of order
21. Notice that $M$ has an automorphism $\phi$ of order 7
which fixes $b$ and sends $a$ to $ab^3.$ A routine
calculation yields that $a (a)\phi (a)\phi^2 (a)\phi^3 = (a)\phi^4.$
We may apply powers of $\phi$ to this equation to deduce that 
there is an epimorphism $\theta: G \longrightarrow M$
defined by $(g_i)\theta = (a)\phi^{i-1}.$

The kernel $H$ of $\theta$ can be shown
to have a presentation on 12 generators and 32 relators. This
can be accomplished via a computer program which uses the
Reidemeister-Schreier method and Tietze transformations.
One can read off from this presentation that $H/H^2$ has order $2^{12}.$
Hence,
by the Theorem of Golod and \v{S}aferevi\v{c} \cite{Jo}, the group  
$H$ is infinite.

These calculations were independently
checked by S. Rees using code of D.~F.~Holt, and by M.~F.~Newman
using SPAS \cite{Fe}, a product of Neub\"user's school at Aachen. 
The SPAS system includes many stand-alone programs written by G.~Havas 
and others.

{\em Added in revision} D~.B.~Nikolova and E.~F.~Robertson \cite{NR} have
recently used related techniques to prove that $F(8,10)$ is infinite.

\begin{thebibliography}{99}
\bibitem{CR} C. M. CAMPBELL and E. F. ROBERTSON, `The orders of certain 
metacyclic groups', {\em Bull. London Math. Soc.} 6 (1974), 312-314. 
\bibitem{Ca}J. J. CANNON, `An introduction to the group theory
language Cayley' in {\em Computational Group Theory}, M.D. Atkinson (ed.), 
Academic Press, London, 1984, 145--183.
\bibitem{Fe} V. FELSCH {\em et al.} , `User's Reference Manual,
SPAS 2.5', Lehrstuhl D f\"ur Mathematik, RWTH  Aachen, (1989).
\bibitem{GN} W. GASCH\"UTZ and M. F. NEWMAN, `On presentations of finite
$p-$groups', {\em J. Reine Angew. Math.} 245 (1970), 172-176.
\bibitem{HR} D. F. HOLT and S. REES,  `A graphics system for displaying 
finite quotients of finitely presented groups' in 
{\em Groups and Computation},
Larry Finkelstein and William M. Kantor (eds.), DIMACS Series in
Discrete Mathematics and Theoretical Computer Science 11, AMS 1993, 113-126.
\bibitem{Jo} D. L. JOHNSON, {\em Presentations of Groups}, 1990, CUP.	
\bibitem{Ne} M. F. NEWMAN,  `Proving a Group Infinite', 
{\em Arch. Math.(Basel)} 54 (1990) No. 3, 209-211.
\bibitem{NR} D. B. NIKOLOVA and E. F. ROBERTSON, `One more infinite
Fibonacci group', {\em C. R. Acad. Bulgare Sci} 46 (1993) No. 3 13-15. 
\bibitem{Th} R. M. THOMAS, `The Fibonacci Groups Revisited',
in C. M. Campbell and E. F. Robertson
(eds.), {\em Proceedings of Groups St. Andrews 1989 }vol. 2 LMS Lecture
Note Series 160, CUP, (1991) pp 445-454.
\end{thebibliography}

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