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\title{The hyperplanes of the $M_{24}$ near polygon}
\author{A.E. Brouwer, H. Cuypers, E.W. Lambeck\\
} 
\maketitle
\def\runninghead{\sc The hyperplanes of the $M_{24}$ near polygon}
\pagestyle{headings}

\begin{abstract}
We give a complete description of the geometric hyperplanes
of the 759-point near hexagon belonging to $M_{24}$.
\end{abstract}

\section{Introduction} 
Consider the unique Steiner system $S(5,8,24)$ on a set $\Omega$ of 24 symbols.
By $\Gamma=(P,L)$ 
we denote the partial linear space (with points set $P$ and line set $L$)
obtained by taking the 759 octads as points, and the triples
of pairwise disjoint octads as lines.
The space $\Gamma$ is the (unique) regular {\em near hexagon} 
of order $(s,t,t_2)=(2,14,2)$
on 759 points; its automorphism group is isomorphic to the Mathieu group $M_{24}$,
see \cite{bcn,bchw,sy}.

A {\em geometric hyperplane}, or hyperplane for short, 
of a partial linear space is a subset of the point set
which meets each line in one or all points of the line.
In this note we determine the $M_{24}$-orbits on the hyperplanes
of $\Gamma$, completing, correcting and simplifying the results of Chapter 4
of \cite{lambeck}.
This classification is used in the study of circular extensions
of $\Gamma$, see \cite{cuypers}.


\section{The hyperplanes of the near hexagon $\Gamma$}


Let $\Gamma=(P,L)$ be the near hexagon on 759 points related to $M_{24}$.
(We use the notation introduced in the previous section.)
In this section we determine the $M_{24}$-orbits on the geometric hyperplanes 
of $\Gamma$. But first we discuss some
results on embeddings of $\Gamma$.

\vspace{2mm}


\noindent
{\bf Projective embeddings.} A ({\em projective}) {\em embedding} of $\Gamma=(P,L)$
is an injective  map $\phi$ of $P$ into the point set of a projective space
$\mathbb{P}$ such that lines are mapped onto lines and $\phi(P)$ generates $\mathbb{P}$. 
The near hexagon $\Gamma$ admits three different 
group-admissible embeddings, i.e., embeddings into a projective space
for which ${\rm Aut}(\Gamma)\simeq M_{24}$ 
acts as a subgroup of the projective 
linear
group on the embedded near hexagon.

First of all, there is the universal embedding into 
$P(U)$, where $U$ is the quotient of $\mathbb{F}_2P$ by the subspace
$K$ spanned by the various vectors  $\Sigma_{x\in l} \ x$, where $l\in L$.
Here a point $p$ is mapped to $p+K$.
By \cite{ronan}, the dimension of $U$ is 23. 
Any other embedding is a quotient of this universal embedding. 

For every point $p\in P$, the points at distance at most 2 from $p$ form
a hyperplane which we denote by $p^\perp$.
The map $p\in P\mapsto p^\perp$ yields an embedding into
the subspace of $P(U^*)$, where $U^*$ is the dual of $U$.
This  subspace is 22-dimensional and the embedding is called
the {\em near polygon embedding}, see \cite{bchw}. 
Finally, we can embed $\Gamma$ into the 11-dimensional Golay code module for $M_{24}$, see \cite{asch}. 
Indeed, if we define the {\em extended binary Golay code} $\cal G$ to be
the subspace of the 24-dimensional space $\mathbb{F}_2\Omega$ generated
by $\Sigma_{x\in B} \ x$, where $B$ runs through the set of octads of the 
Steiner system $S(5,8,24)$, then the {\em Golay code module}
is the space ${\cal G}/\langle v\rangle$ where $v=\Sigma_{x\in \Omega} \ x$,
and the  map $p\in P\mapsto p+\langle v\rangle$
is an embedding of $\Gamma$ into this 11-dimensional Golay code module, see \cite{asch}.


\vspace{2mm}


\noindent
{\bf Hyperplanes.}
The $M_{24}$-orbits on the 
geometric hyperplanes of the near hexagon $\Gamma$ 
are listed in the table below.
We explain the notation of this table and how to find the various
$M_{24}$-orbits on the set  hyperplanes.

Let $H_1$ and $H_2$ be two hyperplanes of $\Gamma$.
Then it is easily seen that the complement $H_1+H_2$
of the symmetric difference of $H_1$ and $H_2$ is also a geometric hyperplane.
So, the hyperplanes form a $\mathbb{F}_2$ vector space, $V$ say, 
which is isomorphic to 
the dual of the universal embedding space $U$ of $\Gamma$, see \cite{ronan}.
Below we will analyze the action of $M_{24}$ on this space of geometric hyperplanes.
In particular, we determine the orbits of $M_{24}$ on this module.
\vspace{2mm}


\noindent
{\bf A $12$-dimensional subspace.}
First we  describe some particular hyperplanes of $\Gamma$, i.e., elements of $V$. 
The trivial hyperplane consisting of all octads
is denoted by $0$, as it is the zero element in $V$. 
If one fixes a symbol $o$ from $\Omega$, 
then each line of $\Gamma$
contains a unique octad containing $o$.
Thus  the set of 253 octads containing this symbol
is a hyperplane without any lines, i.e., it is an {\em ovoid} of $\Gamma$. 
This ovoid will also be denoted by the symbol $o$.
As there are 24 elements in $\Omega$,
this yields 24 hyperplanes.

We will now determine all the hyperplanes 
in the subspace $V_{\cal T}$
of $V$ generated by the 24 symbols in $\Omega$.
The sum of $i$ different  symbols, $i\leq 24$, say $o_1+\cdots +o_i$,
is the geometric hyperplane consisting of the octads missing an even number
of symbols from $o_1, \dots, o_i$.
If $B$ is an octad, then every other octad 
meets $B$ in an even number of points. Thus 
the hyperplane $\Sigma_{o\in B} \ o$ contains every octad and is
equal to $0$.
Since a hyperplane which is the 
sum of at most $7$ symbols is easily seen to be nontrivial,
we find that the submodule of $V$ generated by
the hyperplanes $o$, $o\in \Omega$,
consists of sums of 0, 1, 2, 3 or 4 symbols.
Moreover, since every {\em tetrad} (set of 4 symbols) is in exactly 5 octads,
we can describe a hyperplane, which is the sum of 4 symbols,
in exactly 6 different ways as the sum of 4 symbols. 
The corresponding six tetrads form a partition of $\Omega$, also called a {\em sextet}. We notice that the ${{6}\choose{2}}$ octads that are  the union of two
tetrads in the sextet form a quad in $\Gamma$.
Hence, the group $M_{24}$ has five orbits on $V_{\cal T}$,
of size 1, ${{24}\choose{1}}$, ${{24}\choose{2}}$, ${{24}\choose{3}}$
and ${{1}\over{6}}{{24}\choose{4}}$,
and denoted by $0$, o, oo, ooo and  oooo, respectively.
The submodule $V_{\cal T}$ is isomorphic to the {\em $12$-dimensional Todd
module} for $M_{24}$.
Its  11-dimensional submodule consisting of the hyperplanes
in the orbits $0$, oo and oooo 
is the {\em $11$-dimensional Todd module}, see \cite{asch}. 

\vspace{2mm}


\noindent
{\bf The remaining hyperplanes.}
The near polygon embedding in 22-space is self dual and admits a quotient 
isomorphic to the Golay code module for $M_{24}$.
The kernel of this quotient map is then the 11-dimensional 
Todd module described above. This implies that the 23-dimensional
space of hyperplanes $V$ admits an 11-dimensional quotient
isomorphic to the Golay code module with kernel the 12-dimensional Todd module
$V_{\cal T}$ described above.
 
The group $M_{24}$ has two nontrivial orbits on the Golay code module,
one of size 759 corresponding to the octads of the Steiner system,
and one of size $1288$ corresponding to duums. A {\em duum} is 
a partition of $\Omega$ in two
{\em dodecads} 
(i.e., sets of size 12 that are 
the symmetric difference of two octads meeting in 2 points).
To each octad $B$ we can associate the hyperplane, labeled by $B$,  consisting
of all octads
at distance at most 2 from $B$ in $\Gamma$, i.e.,  octads meeting $B$
in 8, 4 or 0 points.
The $M_{24}$-orbit of
hyperplanes labeled by an octad  is denoted by B. 
If $D$ is a dodecad then each octad meets $D$ in $2$, $4$ or $6$ points.
This implies that either one or all three octads on a line of $\Gamma$
meets $D$ in 4 points.
So to $D$ we can associate the hyperplane, labeled by $D$, 
of all the  octads meeting $D$ in 4 points.
The dodecad $\Omega\setminus D$ determines the same hyperplane as $D$.
The $M_{24}$-orbit of size 1288 on hyperplanes labeled by dodecads
is denoted by D.
The hyperplanes labeled by octads  and dodecads  
are the representatives of the various nontrivial cosets of  $V_{\cal T}$ in $V$.

The set of hyperplanes that are a sum of a hyperplane labeled
by an octad $B$ and 
a hyperplane labeled by up to 3 symbols in or outside
the octad is denoted Bi, Bo, Bii, etc.
Since  the stabilizer of an octad $B$ induces 
the full alternating group on the symbols in $B$ and $2^4:L_4(2)$ on the 16 
symbols outside $B$, we easily see that the group $M_{24}$
is transitive on the sets Bi, Bo$,\dots ,$ Booo.  
If we fix an octad $B$ and a sextet $S$, then $B$ is either the union of
two tetrads of $S$, or it meets  4 tetrads in 2 points, or
it meets each tetrad from $S$ in one or three points.
In other words, the point $B$ of $\Gamma$ is either in the quad of $\Gamma$ determined by the sextet $S$, at distance 1 or at distance 2 from this quad. 
This leads to three $M_{24}$-orbits on hyperplanes denoted by Biiii, Biioo and Biiio, respectively. 

The set of hyperplanes that are a sum of a hyperplane labeled
by a dodecad $D$ and 
a hyperplane labeled by up to 3 symbols in or outside
the dodecad is denoted Di, Do, Dii, etc.
The stabilizer of a dodecad $D$ induces 
the group $M_{12}$ on $D$ in one of its 5-transitive actions. 
On the complement of $D$
it also induces  the group $M_{12}$ on $D$ in one of its 5-transitive actions,
but the two actions on $D$ and its complement are not the same.

From this one easily derives  
that the group $M_{24}$
is transitive on the sets Di, Do$,\dots,$ Diio.  
(Notice that by interchanging the r\^ole of $D$ and its complement we may assume
that the majority of the symbols is inside $D$.)
It remains to describe the $M_{24}$-orbits on  hyperplanes that are
sums of hyperplanes labeled by  a  dodecad and hyperplanes labeled
by four symbols.
Fix a dodecad $D$.  Then have one of the following situations for a sextet $S$.
One tetrad of $S$ is in $D$, one misses $D$ and the other four
meet $D$ in two points; since the stabilizer of $D$ induces the 5-transitive
group $M_{12}$ on $D$, this leads to an orbit on hyperplanes of size
$1288 \times {{12}\choose{4}}$ denoted by Diiii.
There are  three  tetrads in $S$ meeting $D$ in 3 points and three
meeting it in 1 point. This leads to an orbit of hyperplanes  of size
$ {{12}\choose{3}}\times 12\times {{1}\over{3}} \times 1288$  denoted by Diiio.
Finally, the remaining 396 sextets $S$ only have tetrads
meeting $D$ in 2 points.
This leads to an
$M_{24}$-orbit of hyperplanes of size $396 \times 1288$ denoted by Diioo.

\vspace{2mm}


\noindent
{\bf The table.}
In the table below, we give for each $M_{24}$-orbit on the hyperplanes,
its name (explained above), the universal embedding dimension
of the partial linear space induced on a hyperplane in the orbit
(found by computer),
the size of the hyperplanes in the orbit (viewed as collection of octads),
the size of the orbit, and the {\em line distribution} of a hyperplane
$H$ in the orbit: in $i^{a_i}$
the number $a_i$ is the number of points of $H$ that
are on precisely $i$ lines entirely contained within $H$.
Moreover the name used in \cite{lambeck} is also given in the column 
EWL name. Under the header submodule we describe in which submodule
an orbit can be found,
the 11-dimensional Todd module 11-{\em Todd}, the 12-dimensional Todd module 
12-{\em Todd}, the near polygon embedding module {\em Near Polygon} and the dual universal space 
{\em Universal}.
(Notice that 11-Todd is contained in 12-Todd and Near Polygon and that
all are contained in Universal.)
Since the orbit sizes add up to the right number $2^{23}$,
the list is complete. The line distribution helps to identify
a given orbit with one from the list.

\vspace{3mm}



\begin{center}
{\small
$M_{24}$-orbits on the hyperplanes of $\Gamma$

\vspace{3mm}

\begin{tabular}{|c c c c c c l|}
\hline
EWL name&  name&   udim&  size&   orbit size& submodule &line distribution\\
\hline
(X)&    $0$&    23&   759&    1&      11-Todd&   $15^{759}$\\
(H13)&  oo&     22&   407&    276&    11-Todd&   $7^{330} 15^{77}$\\
(H4)&   oooo&   23&   375&    1771&   11-Todd&   $7^{360}15^{15}$\\
&       o&      253&  253&    24&     12-Todd&   $0^{253}$\\
&       ooo&    43&   381&    2024&   12-Todd&   $0^{21} 8^{360}$\\

(H1)&   B&      51&   311&  $1 \times 759$& Near Polygon&$3^{280} 15^{31}$\\
(H14)&  Bii&    23&   343& $28 \times 759$& Near Polygon&$3^{60} 5^{192} 7^{30} 11^{60} 15^1$\\
(H7)&   Bio&    23&   351&$128 \times 759$& Near Polygon&$3^{35} 5^{126} 7^{120} 9^{70}$\\
(H10)&  Boo&    23&   407&$120 \times 759$& Near Polygon&$3^{14} 7^{120} 9^{224} 11^{42} 15^{7}$\\
(H2)&   Biiii&  23&   439& $35 \times 759$& Near Polygon&$3^8 7^{24} 9^{256} 11^{144} 15^7$\\
(H9)&   Biiio&  23&   383&$896 \times 759$& Near Polygon&$3^5 5^{78} 7^{120} 9^{150} 11^{30}$\\
(H5)&   Biioo&  23&   375&$840 \times 759$& Near Polygon&$3^6 5^{96} 7^{126} 9^{128} 11^{18} 15^1$\\
&       D&      22&   495&  $1 \times 1288$&Near Polygon&$11^{495}$\\
(H3)&   Dii&    23&   367&$132 \times 1288$&Near Polygon&$3^{30} 5^{72} 7^{180} 9^{40} 11^{45}$\\
(H11)&  Dio&    23&   407&$144 \times 1288$&Near Polygon&$5^{22} 7^{165} 9^{110} 11^{110}$\\
(H12)&  Diiii&  23&   367&$495 \times 1288$&Near Polygon&$3^{24} 5^{64} 7^{192} 9^{64} 11^{23}$\\
(H6)&   Diiio&  23&   375&$880 \times 1288$&Near Polygon&$3^{18} 5^{54} 7^{189} 9^{78} 11^{36}$\\
(H8)&   Diioo&  23&   399&$396 \times 1288$&Near Polygon&$5^{24} 7^{180} 9^{120} 11^{75}$\\


&Bi&    23& 477&    $8 \times 759$  &Universal&$0^1 10^{336} 12^{140}$\\
&Bo&    37& 365&    $16 \times 759$ &Universal&$0^{15} 6^{280} 12^{70}$\\
&Biii&  24& 349&    $56 \times 759$ &Universal&$0^1 4^{120} 6^{160} 10^{48} 12^{20}$\\
&Biio&  23& 397&    $448 \times 759$&Universal&$4^{15} 6^{100} 8^{135} 10^{132} 12^{15}$\\
&Bioo&  23& 381&    $960 \times 759$&Universal&$4^{21} 6^{140} 8^{129} 10^{84} 12^7$\\
&Booo&  26& 365&    $560 \times 759$&Universal&$0^3 4^{36} 6^{168} 8^{108} 10^{48} 12^2$\\
&Di&    23& 341&    $24 \times 1288$&Universal&$4^{165} 6^{110} 10^{66}$\\
&Diii&  23& 405&    $440 \times 1288$&Universal&$4^9 6^{54} 8^{216} 10^{90} 12^{36}$\\
&Diio&  23& 373&    $1584 \times 1288$&Universal&$4^{55} 6^{90} 8^{180} 10^{38} 12^{10}$\\

\hline
\end{tabular}
}

\end{center}

\vspace{3mm}

\noindent
{\bf Subspaces.}
A subset $S$ of a partial linear space is called {\em subspace}
if it contains every line that meets it in at least two points.
The partial linear space $\Gamma$ has many subspaces, and one may
wonder whether any reasonable classification is possible.

A way to obtain many subspaces is the following:
Let $G$ be an abelian group, and $u = (u_\omega)$ a vector
indexed by $\Omega$ with elements in $G$ such that $\sum u_\omega = 0$.
Then $S(u) := \{B | \sum_{\omega \in B} u_\omega = 0\}$ is a subspace.
If we call such subspaces {\em abelian}, then the intersection
of two abelian subspaces is again abelian.

For example, let us take $G = \Z_{16}$, then, in obvious notation,
we have: oo $= S(8^2 0^{22})$, B $= S(4^8 0^{16})$,
D $= S(4^{12} 0^{12})$, o $= S(7^1 (-1)^{23})$.
We see that there is an embedding (of abelian groups) of $V$
into the quotient of $\Z_{16}^{24}$ by the subgroups spanned by
twice the all-1 vector and 8 times the extended binary Golay code;
in the image all coordinates are congruent mod 4 and sum to zero.

A few more large subspaces (that are not hyperplanes) are found
in the same way. For example, the subspace $S(u)$ where
$u = (2^4 (-2)^4 0^{16})$ (with the $4+4$ positions forming an octad)
is a subspace on 431 points and udim 22, contained in the hyperplane
$S(2u)$ of type Biiii with 439 points. Similarly, the subspace $S(u)$
with $u = (1^{11} (-3)^1 (-1)^{11} 3^1)$ (with the $11+1$ positions forming
a dodecad) is a subspace on 385 points and udim 22 contained in the
hyperplane $S(2u)$ of type Dio with 407 points.

In this way it happens that most hyperplanes have udim 23 again---they
are of the form $S(2u)$ and have hyperplanes $S(u)$ not obtained by
intersection with a hyperplane in the entire space. (Note that $u$
is not determined by $2u$.)
Of course, isolated points, visible in the line distribution
as points on zero lines, each add 1 to udim. Finally hyperplanes
of type B are unions of a bouquet of 35 quads on a point, and
visibly have udim $1+15+35=51$.

\vspace{2mm}

\noindent
{\bf Spanning.}
A subset $A$ is said to {\em span} a subspace $S$ when $S$ is
the smallest subspace containing $A$. Clearly, this implies
that $|A| \ge {\rm udim}(S)$.
Cooperstein asked whether $\Gamma$ is spanned by 23 points,
and this is indeed the case, as computer calculation reveals.
More generally, each of the hyperplanes $H$ listed in the table
can be spanned by ${\rm udim}(H)$ points. Of course there do exist
partial linear spaces $S$ with lines of size 3, for which one needs
more than ${\rm udim}(S)$ points to span. For example, the affine
plane $AG(2,3)$ on 9 points is spanned by 3 points but has no
hyperplanes so that its udim is 0.

A referee asked for a less trivial example. One can manufacture
a small partial linear space $S$ with ${\rm udim}(S) = 6$
that requires 7 points to span as follows. Let a {\em tripod}
with {\em feet} $p,q,r$ be a set of seven points, say
$x,a,b,c,p,q,r$, and four lines, namely $xap, xbq, xcr, abc$.
Let $S$ be the partial linear space with 21 points and 15 lines
obtained by taking the union of three tripods with feet
$p_i,q_i,r_i$ ($i=1,2,3$) and adding the three lines
$p_1p_2p_3, q_1q_2q_3, r_1r_2r_3$.
Now $S$ has an injective embedding into a projective space.

{\small 
\renewcommand{\theenumi}{\arabic{enumi}}
\noindent 
\begin{thebibliography}{xx} 
\bibitem{asch} M. Aschbacher, {\em Sporadic simple groups},
Cambridge tracts in Mathematics {\bf 104}, Cambridge University Press, Cambridge, 1994. 


\bibitem{bcn} A.E. Brouwer, A.M. Cohen and A. Neumaier, {\it Distance-regular graphs},
Ergebnisse der Math., Springer Verlag, Heidelberg, 1989.

\bibitem{bchw} A.E. Brouwer, A.M. Cohen, J.I. Hall and H.A. Wilbrink, Near polygons with lines 
of size three and Fischer spaces, {\em Geom. Dedicata}
{\bf 49} (1994), 
349--368.

\bibitem{cuypers} H. Cuypers,
{\em Extended near hexagons and line systems}, 
preprint, Eindhoven University of Technology, 1999.


\bibitem{lambeck} E.W. Lambeck, {\em Contributions to the theory of distance regular graphs},
Thesis, Eindhoven University of Technology, 1990.



\bibitem{ronan}M. Ronan, Embeddings and hyperplanes of discrete
geometries, {\em European J. Combin.} {\bf 8} (1987), 385--406. 

\bibitem{sy} E.E. Shult and A. Yanushka, Near polygons and line systems,
{\em Geom. Dedicata} {\bf 9} (1980), 1--72.


\end{thebibliography}



\vspace{.5cm}

\begin{minipage}{8cm}
Andries E. Brouwer and 
Hans Cuypers\\
Department of Mathematics\\ 
Eindhoven University of Technology\\ 
P.O. Box 513, 5600 MB Eindhoven\\ 
The Netherlands\\
\end{minipage}
\begin{minipage}{8cm}
Ernst W. Lambeck\\
Ouderf 3\\
4824 HV Breda\\
The Netherlands\\
\phantom{a}\\
\end{minipage}
}
\end{document}
