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\topmatter 
\title A Coordinatization of Generalized Quadranlges of Order
	  $(s-1,s+1)$
\endtitle 
\author Marijke de Soete
\endauthor 
\address
Seminarie voor hogere Meetkunde,
State University of Ghent, Krijgslaan 281, B-9000 Gent, Belguim
\endaddress

\endtopmatter
\document
A generalized quadrangle is an incidence structure $S=(P,B,I)$ with
$P$ and $B$ sets of objects called points and lines respectively, with
a symmetric incidence relation $I$ which satisfies:

\roster 
\item"(i)" each point is incident with $1+t$ lines ($t\ge1$) and two
distinct points are incident with at most one line;
\item"(ii)" each line is incident with $1+s$ points ($s\ge1$) and two
distinct lines are incident with at most one point;
\item"(iii)" for each point $x$ and each line $L$, $x\!\not\! I L$,
there exists a unique pair $(y,M)\in P\times B$ such that
$x\,I\,M\,I\,y\,I\,L $.
\endroster

We call $(s,t)$ the order of $S$.

Let us consider a generalized quadrangle $S$ of order $(s-1,s+1)$
containing a spread $R$ (i.e., a subset $R$ of $B$ such that each
point is incident with a unique line of $R$). $R$ is called a spread
of symmetry for the generalized quadrangle $S$ if the group $G_R$ of
automorphisms of $S$ fixing $R$ linewise acts transitively on each
line of $R$. If $S$ has a spread of symmetry, then from $S$ there
arises a generalized quadrangle $S'$ of order $s$ having a center of
symmetry. So in this case we are able to give a coordinatization of
$S$, using a planar ternary ring, and $G_R$, which is derived from the
coordinatization of $S'$ due to S.~E.~Payne \cite{3}.

We also investigate the converse problem. Given a planar ternary ring
and a group $G$, which are the conditions to obtain a generazlied
quadrangle of order $(s-1,s+1)$. Examples are given for the known
models $AS(q)$, $T^*_2(O)$ and the dual of $P(T_2(O),x)$ with $x$ a
point of the oval $O$.


\Refs
\ref\no 1\by M. De Soete,  J. A. Thas\paper Acoordinatization of 
generalized quadrangles of order $(s,s+2)$\jour J. Combin\. Theory 
Ser.~A\vol 48\yr 1988
\pages 1--11\endref
\ref\no 2\by Stanley E. Payne\paper Quadrangles of order $(s-1,\,s+1)$
\jour J. Algebra \vol 22
\yr 1972\pages 97--119\endref
\ref\no 3\by Stanley E. Payne\paper  Generalized quadrangles of 
even order\jour J. Algebra\vol 31
\yr 1974\pages 367--391\endref
\endRefs



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