# Introduction

This chapter presents the functions designed for constructing and computing with incidence geometries and coset geometries.

We recall the basic definitions and notation in the field of Incidence Geometry. We refer to the *Handbook of Incidence Geometry* [[Buekenhout, 1995](../../references.md#cite-buekenhout95)], edited by Francis Buekenhout, or to Antonio Pasini’s book *Diagram Geometries* [[Pasini, 1994](../../references.md#cite-pasini94)], for a more detailed overview of the subject.

Let $X$ and $I$ be two finite sets. Let $t : X \rightarrow I$ be a mapping from $X$ onto $I$. Let $\sim$ be a reflexive and symmetric relation such that $\forall x,y \in X$, $x \sim y$ and $t(x) = t(y) \Rightarrow x = y$. The four-tuple $\Gamma(X, \sim, t, I)$ is what we call an *Incidence Geometry* in Magma. Remark that it is not a geometry in the sense of Buekenhout since we do not impose that every flag (i.e. clique of the incidence graph) of $\Gamma$ must be contained in a chamber (i.e. a clique containing one element of each type). If the latter condition is satisfied, then an incidence geometry is a geometry in the sense of Buekenhout. The set $X$ contains the *elements* of the geometry, while $I$ is called the set of *types*. The function $t$ is called the *type function* and $\sim$ is called the *incidence relation* of $\Gamma$. The cardinality of $I$ is the *rank* of $\Gamma$.

It is possible to construct incidence geometries from a group and some of its subgroups using an algorithm first introduced by Jacques Tits in 1962 [[Tits, 1962](../../references.md#cite-tits1962)]. Let $G$ be a group and let $I$ be a finite set. Let $\lbrace G_i, i \in I\rbrace$ be a set of subgroups of $G$. Define $X = \lbrace G_ig, g\in G, i\in I\rbrace$ to be the set of elements of $\Gamma$. Define the type function as $t : X \rightarrow I : G_ig \rightarrow i$ and the incidence relation as follows: $G_ig \sim G_jh$ iff $G_ig \cap G_jh \neq \emptyset$. The subgroups $\lbrace G_i, i \in I\rbrace$ are called the *maximal parabolic subgroups*. The subgroup $\cap_{i\in I}G_i$ is called the *Borel subgroup*. Finally, the subgroups $\lbrace \cap_{j\in I\setminus\{i\}}G_j , i \in I\rbrace$ are called the *minimal parabolic subgroups*. These geometries are called *Coset Geometries* in Magma to remind the user that they are constructed from a group. Again, a coset geometry is not a geometry in the sense of Buekenhout. If every flag of the coset geometry is contained in a chamber, then it is a Buekenhout geometry. We will see that, using coset geometries, it is easy to build huge incidence geometries by giving very little data.

The category names for the incidence geometries and coset geometries are:

- Incidence Geometry : `IncGeom`

- Coset Geometry : `CosetGeom`