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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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