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\title{A construction of the Suzuki graph}
\author{A.E. Brouwer}
\date{2008-07-08}
\maketitle

\begin{abstract}
We give an elementary construction of the rank 3 Suzuki graph,
following Horiguchi, Kitazume \& Nakasora.
\end{abstract}

\section{Introduction}
The Suzuki sporadic simple group Suz was discovered by M. Suzuki \cite{suz}
as index 2 subgroup of the automorphism group of a rank 3 graph $\Sigma$,
strongly regular with parameters $(v,k,\lambda,\mu) = (1782,416,100,96)$,
the largest graph in the Suzuki tower (of graphs for Suz:2,
$G_2(4)$:2, HJ:2, $U_3(3)$:2, $L_3(2)$:2).
The spectrum of $\Sigma$ is $416^1$, $20^{780}$, $(-16)^{1001}$.

The Hoffman bound for the size of a coclique in $\Sigma$ is 66,
and in fact this graph has a single orbit of cocliques of this size,
as was found by Kuzuta \cite{kuzuta} and is easy to check.

If $C$ is a 66-coclique in $\Sigma$, then each point outside $C$
has 16 neighbours in $C$, so that the 1716 points outside $C$
induce the structure of a block design with parameters
2-(66,16,96) on $C$. It was observed by Horiguchi, Kitazume \& Nakasora
\cite{kita}, that this block design happens to be a 3-(66,16,21).

Something funny happens here: usually when one has a maximal coclique
in a graph belonging to a nice group, then the group is transitive
on the vertices of the coclique. But here the stabilizer of $C$ in
Suz:2 is $U_3(4)$:4 acting with vertex orbit sizes $1 + 65$,
2-transitively on the orbit of length 65.

In \cite{kita} the graph $\Sigma$ was constructed in this setting,
starting from designs on the unital in $PG(2,16)$.
We repeat their construction here, perhaps in a more elegant way.

\section{A 2-(65,15,21) design and the $G_2(4)$ graph}
Consider the vector space $V$ of dimension 3 over the field $\Ff_{16}$,
provided with a nondegenerate Hermitean form $\langle \cdot,\cdot \rangle$.
The projective plane $PV$ has 65 isotropic and 208 nonisotropic points.
There are $208.12.1/6 = 416$ orthogonal bases (triples of orthogonal
nonisotropic points).

We would like to construct the local graph $\Gamma$ of $\Sigma$,
a strongly regular graph with parameters $(v,k,\lambda,\mu) =
(416,100,36,20)$. The group $U_3(4)$:4 of semilinear transformations
preserving the form acts transitively on the 416 bases, with rank 5.
The suborbit sizes are 1, 15, 100, 150, 150.
The graph $\Gamma$ is the graph on the 416 bases obtained by taking
the suborbit of size 100 for adjacency.

These suborbits can be described geometrically as follows:
Given one basis $\{a,b,c\}$, the suborbit of size 15 consists of
the bases that have an element in common with $\{a,b,c\}$.
The first suborbit of size 150 consists of the bases that are disjoint
from $\{a,b,c\}$ but contain a point orthogonal to one of $a,b,c$.
Associated with a basis $\{a,b,c\}$ is the triangle consisting of
the 15 isotropic points on the three lines $ab$, $ac$, and $bc$.
The suborbits of sizes 1, 15, 100, 150, 150 correspond to the bases
with triangles having 15, 5, 3, 2, 5 points in common, respectively.
Thus, $\Gamma$ can be described as the graph on the 416 triangles,
adjacent when they have 3 points in common (\cite{kita}).

These 416 triangles form a 2-(65,15,21) design.
% (More generally, in $PG(2,q^2)$ with unital, these triangles
% form a 2-$(q^3+1,3(q+1),\frac{1}{2}(q-1)(3q+2))$ design.)

\section{A 3-(66,16,21) design and the Suz graph}
The set of 16 hyperbolic lines on any isotropic point $y$ carries
the structure of an affine plane of order 4.
This affine plane has 20 lines of size 4, so we find 20 sets of four lines
on $y$, and taking only the isotropic points distinct from $y$ on these
lines, we find for each $y$ 20 sets of 16 isotropic points, and
altogether 1300 such sets. Since $U_3(4)$:4 acts 2-transitively
these 1300 sets will be the blocks of a 2-design with parameters
2-(65,16,75).

If we add a point $\infty$ to the set of isotropic points, and to the
blocks (triangles) of the 2-(65,15,21) design found earlier, then the
union of both designs is clearly a 2-(66,16,96), and in fact even a
3-(66,16,21) design. Indeed, the group is transitive on collinear
triples of isotropic points and on noncollinear such triples
so only a single equality needs to be checked.

Let $(X,{\cal B})$ be the 3-(66,16,21) design just constructed,
with 66 points and 1716 blocks.
It has block intersection numbers 0, 3, 4, 6, 16, and each block
meets $n_i$ blocks in $i$ points,
where $n_0 = 75$, $n_3 = 800$, $n_4 = 400$, $n_6 = 440$, $n_{16} = 1$.

Now construct the Suz graph with $1782 = 66+1716 = 1+65+416+1300$ vertices,
where the $1+65$ is a 66-coclique, the 416 are the neighbours of $1$
and the $416+1300$ (viewed as blocks) are each adjacent to their 16 points.
We already saw that "having 3 points in common" was the right adjacency
for the 15-point triangles, and now that $\infty$ has been added
the construction becomes: let two blocks be adjacent when they have
4 points in common. This defines the sporadic Suzuki graph.


\begin{thebibliography}{99}

\bibitem{kita}
N. Horiguchi, M. Kitazume \& H. Nakasora,
{\em A construction of the sporadic Suzuki graph from $U_3(4)$},
preprint, 2008.

\bibitem{kuzuta}
K. Kuzuta, Master Thesis, Chiba Univ., 2007 (in Japanese).

\bibitem{suz}
M. Suzuki,
{\em A simple group of order 448,345,497,600},
pp. 113--119 in: Theory of Finite Groups
(Symposium, Harvard Univ., Cambridge, Mass, 1968),
Benjamin, New York, 1969.

\end{thebibliography}

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