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First we cite some important references:
\cite{Buek79,BuCD98,Buek95,hbk,BuCD97,BuDL99,BiVN36}
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As the theorem \cite[p. 97]{DiMo98} says, we have to take into account the isomorphism problem.
The goal of this work is to study incidence geometries on which a finite
symmetric or alternating group acts flag-transitively. \label{start}
The idea of incidence geometry arises from the structure induced on the set
of points and lines of elementary geometry by the inclusion relation. When a
point belongs to a line, one often says that the line is
\emph{incident}
with that point. In the same way, a point can be said to be incident
with a line if the line contains the point. The separation of the
incidence structure from the betweenness, congruence and ``continuity''
was made by \textsc{Hilbert} in his famous \emph{Grundlagen der Geometrie}
\cite{Hilb1899}. This \emph{incidence relation} on the set of
points and lines of the Euclidean plane can then be seen as a
kind of ``symmetrisation'' of the inclusion of points in lines. Working
in this way, we also obtain symmetry among the objects of elementary
geometry. Points and lines can be treated in the same manner, even if there
is no duality around. This is generalized in the field of mathematics which
is called \emph{Incidence Geometry.} This branch made a lot of progress
during
the last 20 years. An account of this progress together with most of the
theoretical background is given in the \emph{Handbook of Incidence
Geometry}~\cite{hbk} edited by \textsc{Buekenhout}.
An \emph{incidence geometry} can be seen as a mathematical object consisting
of a set, whose elements are
partitioned in different \emph{types}, for example point, line,
hyperplane,\,\ldots\ or table, chair, beermug,\,\ldots\ (like
\textsc{Hilbert}
suggested it), with a symmetric \index{incidence relation}\emph{incidence
relation} given on the
elements of this set. The number of different types is called the \emph{rank}
of the incidence geometry. For precise definitions, the reader is referred to
section~\ref{sec:incgeo}.
The study of incidence geometries leads to a dialogue between geometry and
group theory, in the spirit of \textsc{Klein}'s famous
\emph{Erlanger Programm} of 1872. In fact, the way to associate a group with
a
given incidence geometry is known since \textsc{Klein}: one can take the
group
of automorphisms of the geometry. Assuming that the automorphism group acts
flag-transitively, \textsc{Tits} even went further and
completely translated the elements, types and incidence relation into the
language of groups, bringing the geometry back to a group together with a
collection of subgroups (see theorem~\ref{trm:cosetrep}). To construct a
flag-transitive
incidence geometry from a group, we need more than the group alone. We need
a group together with a collection of subgroups satisfying certain
conditions. Experience shows that certain requirements are not very
restrictive so that a lot of
geometries can be obtained with them. Once a
choice of subgroups of a given group has been made, the construction of the
geometry is ! Package footbib Error: the output routine of LaTeX changed.
algorithmic and can be programmed in a computer algebra package such as
\textsf{GAP}~\cite{GAP:manual} or \textsf{MAGMA}~\cite{BCaP97}. The algorithm
we are talking about is
called the ``Tits-algorithm'', in honor of its inventor, Jacques
\textsc{Tits}
(see page~\pageref{titsalgo}). The resulting geometry can be seen as a
geometric representation of the given group and can be used to study that
group. Some choices of subgroups yield more useful geometries than others. A
success story in this direction is the \emph{Theory of
Buildings}~\cite{Tits74} developed by \textsc{Tits} in the 60s.
Many groups have a natural geometric interpretation. For instance,
the so called
\emph{Classical Groups}, which are related to
automorphisms of vector
spaces and
projective spaces. There is a generalization of these classical groups:
simple groups of Lie type or Lie groups for short. The study of all complex
simple Lie groups leads to a classification of these. It turns out that only
five of
them are not classical groups. These are called \emph{exceptional groups}. The
quest for a geometric representation of these groups led \textsc{Tits} to the
invention of the fruitful \emph{Theory of Buildings}. By selecting particular
subgroups of the simple Lie
groups and applying the Tits-algorithm, one obtains an incidence geometry
which is suitable to explain the group. Buildings are incidence
geometries, which now help us to understand the simple Lie groups better than
before. They were for instance used as a recognition tool for the finite
simple groups of Lie type in the \emph{classification of the finite simple
groups}.
The
classification of finite simple groups~\cite{CFSG} is one of the major
achievements in mathematics and results from many efforts done in
the areas of algebra and geometry. The theorem is that a nonabelian finite
simple group must be one of the following: a simple Lie
group, an alternating group or one of 26 isolated cases called
\emph{sporadic
groups.} An important question is to find a
geometric interpretation for these groups. We are in a situation similar to
the one \textsc{Tits} was facing in the 50s. Part of the question is
answered by the theory of buildings and the exceptions are the alternating
groups and the 26 sporadic groups. A lot of work has been done for
sporadic simple groups (see chapter~22 of~\cite{hbk} and~\cite{Asch94b} for
instance).
One must remark that for each simple group of Lie type, there is a
\emph{unique} building describing it (except for some special cases where we
have two buildings resulting from a sporadic isomorphism).
Inspired by this, we could think (or dream) of a new
theory, including the buildings and associating ``building-like'' geometries
with groups from a broader class. Suggestions for this have been made by
\textsc{Buekenhout} in \cite{Buek95}. The idea is to restrict the choice of
the subgroups to use in the Tits algorithm in such a way that we get a small
number (preferably \emph{one}) of incidence geometries for a given group in
the class considered. A good class to start with is that of the nonabelian
finite simple groups.
Experiments on small groups to which I participated (see for
instance~\cite{BuCD98}) inspire some conditions
which restrict the
number of resulting geometries. Successful
properties are WPRI, RWPRI, PRI and RPRI, which are related to primitive
actions of groups. They will be used in this work and are defined in
section~\ref{sec:gpgeo}.
Another property which also diminishes the number of geometries is $(IP)_2$
(defined on page~\pageref{defi:ip2})
which is a generalization of the well-known Euclidean assertion that ``there
is only one line passing through two distinct points''.
In the present work I concentrate on the RWPRI geometries on which a
symmetric group acts flag-transitively. An objective is to find infinite
families of such geometries as a step towards a classification.
It is well-known that
primitivity of an action is closely related to maximality of
point-stabilizers. Therefore the classification of RWPRI geometries will use
a classification of the maximal
subgroups of the alternating and symmetric groups (see~\cite{LiPS87} for
example). If we take a
maximal subgroup of a symmetric group $Sym(S)$, it can be either
transitive or
intransitive. In the latter case we have the stabilizer of a subset of
$S$. The transitive case splits in primitive and imprimitive
action. Imprimitive actions correspond to wreath products and the
primitive case can be divided in families by the O'Nan-Scott
theorem (see for example~\cite{Came99,DiMo96}). These families are under partial control
by the
classification of finite simple groups.
In this thesis, I study the intransitive case.
For the symmetric groups of degree less than~7, the RWPRI geometries were
computed and can be found in~\cite{BuDL99}. Observation of the geometries in
this ``atlas'' of geometries inspired me and yields large family of
flag-transitive RWPRI geometries which is the exclusive subject of
chapter~\ref{chap:img}. The geometries of this family are called
\emph{inductively minimal geometries} and were introduced by
\textsc{Buekenhout}, \textsc{Dehon} and myself in~\cite{BuCD97}.
These geometries are completely
classified, a rather unexpected result because of the large
family of Coxeter diagrams involved. The diagrams involved are called
\emph{IM-diagrams} and are described in section~\ref{sec:imdiag}. It is shown
that there is a one-to-one correspondence between inductively minimal
geometries and IM-diagrams. This is achieved by proving that the
diagram of
an inductively minimal geometry must be an IM-diagram and that we can
construct a unique inductively minimal from any given IM-diagram. I also
solve the enumeration of inductively minimal geometries by producing a
bijection between the class of trees on $n+1$ vertices and the
the class IM-diagrams on $n$ vertices (see section~\ref{sec:imgenum}).
Further in chapter~\ref{chap:img}, I study the truncations of inductively
minimal geometries which satisfy $(IP)_2$ and RWPRI. They are characterized
by means of their diagram in section~\ref{sec:timg}. An unexpected result is
that every $(IP)_2$ truncation of an inductively minimal geometry
automatically satisfies the RWPRI property.
From inductively minimal geometries I obtain a
new infinite family of Petersen
geometries. These are geometries having a residue isomorphic to
the vertex-edge-system of the Petersen graph (see chapter~\ref{chap:prereq}
for the vocabulary used here).
In general, Petersen geometries have been found useful in
the explanation of many
sporadic groups (see~\cite{Ivan99}) and are studied by several people. The
infinite families known up till now (see chapter~22 of~\cite{hbk}) all have a
linear diagram but the new family is more general and provides plenty of
Petersen geometries with nonlinear
diagram. Moreover, it generalizes one of the
known families.
These Petersen geometries appear in chapter~\ref{chap:petgeo} and are
called \emph{Petersen replacements}. The main result of that chapter can be
phrased as follows:
\begin{quote}
\itshape
Given an inductively minimal geometry and a particular connected component
$I_1$ of its diagram $(I,\sim )$, we can construct a
Petersen geometry whose diagram is
obtained by replacing the
component $I\setminus I_1$ by a Petersen string diagram of appropriate
length. The symmetric and alternating group of degree $|I|+1$
both act flag-transitively on this new geometry which also satisfies
the RWPRI and (IP)$_2$ properties.
\end{quote}
By the way: the front cover of this thesis shows an inductively minimal
geometry trying to catch a Petersen string for a replacement.
A spectacular fact is that 119 of the 153
RWPRI geometries listed in~\cite{BuDL99} for the symmetric groups
are inductively minimal geometries, Petersen replacements
or truncations of these. This is more than 3/4 of the geometries!
To see if the success of inductively minimal geometries is confirmed for
higher degrees, I decided to construct the RWPRI geometries for the
symmetric group of degree~8, the first group which was not accessible by
computer at that time (1997). This was done interactively with the computer
using our knowledge of the subgroup lattice of $\Sym{8}$ (which can be found
in~\cite{Mill96}). Due to the high complexity of finding the maximal
subgroups of maximal subgroups in $\Sym{8}$, the computations were only
completed up to rank~3. This also shows that a general theoretical treatment
using (classifications of) maximal subgroups of the symmetric groups and
their maximal subgroups is not in reach for the time being. However, this
method was applied successfully by \textsc{Leemans} for the Suzuki simple
groups
which have a more accessible subgroup structure (see~\cite{Leem98}). In the
meantime
computer power has increased and I could compute all
flag-transitive RWPRI geometries for the alternating group of degree~8 in
about 13 days of CPU time. This list verifies the work done for $\Sym{8}$ in
low rank. Chapter~\ref{chap:list} contains a list of all RWPRI geometries for
$\Alt{8}$ and states some general theorems explaining the geometries related
to inductively minimal geometries. These theorems take care of 68
of the 94 geometries in the list. For all remaining
geometries, I provide a geometric construction proving their existence. These
constructions use objects related to the projective space $PG(3,2)$ and the
Klein correspondence with the projective space in dimension 5. This is
a consequence of the sporadic isomorphisms
$A_8\cong PSL(4,2)\cong PO^+_6(2)$.
Another piece of general theory studies the relation between RWPRI geometries
and independent generating sets. A set $\{ g_1,
g_2,\ldots ,g_n\}$ of elements in a group is called \emph{independent} if
for every $i\in I=\{ 1,2,\ldots ,n\}$ we have $g_i\not\in\langle g_j\mid
j\in I\setminus\{ i\}\rangle$. In chapter~\ref{chap:indgens}, I prove
that an RWPRI geometry on whose chambers a
group $G$ acts regularly, yields an
independent set which generates $G$. In the proof, the property RWPRI plays a
crucial role, hinting at the importance of this property
and promising deeper results. However it is not true that an independent
generating set automatically gives a geometry. A pregeometry can be obtained
by applying the \emph{Tits algorithm}
and I give additional conditions turning this into
a geometry. I also study the independent sets of transpositions which
generate a symmetric group. Using trees again I characterize these sets.
An important result of the theory developed in chapter~\ref{chap:indgens} is
that the highest possible rank for an RWPRI geometry on which the symmetric
group of degree $n$ acts flag-transitively is $n-1$. That this bound is
tight, is shown by the inductively minimal geometries for $\Sym{n}$ which
have precisely rank $n-1$. This result uses a theorem by
\textsc{Whiston}, saying that an independent generating set of $\Sym{n}$ has
at most $n-1$ elements (see theorem~\ref{trm:Whiston}). This interaction
between
geometry and the theory of permutation groups promises more results in the
future.
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