# Properties of Incidence Geometries and Coset Geometries

Let us recall definitions of the properties described in this section.

An incidence geometry $\Gamma$ is *flag–transitive* if for every two flags $x,y$ of the same type of $\Gamma$, there exists an element $g$ of $Aut(\Gamma)$ such that $g(x) = y$. We also say that $Aut(\Gamma)$ acts flag–transitively in this case.

Moreover, it is a flag–transitive geometry if it contains at least one chamber.

A coset geometry $\Gamma(G; (G_i)_{i\in I})$ is *flag–transitive* if for every two flags $x,y$ of the same type of $\Gamma$, there exists an element $g$ of $G$ such that $g(x) = y$. It is then a flag–transitive geometry since the set $\lbrace (G_i)_{i\in I} \rbrace$ is a chamber of $\Gamma$.

## `IsFTGeometry(D): IncGeom -> BoolElt`

Given an incidence geometry $D$, return `true` if and only if the automorphism group of $D$ acts flag–transitively on $D$ and $D$ has at least one chamber.

## `IsFTGeometry(C): CosetGeom -> BoolElt`

Given a coset geometry $C$, return `true` if and only if the group of $C$ acts flag–transitively on $C$.

## `IsFirm(X): IncGeom -> BoolElt`

## `IsFirm(X): CosetGeom -> BoolElt`

Given either a coset geometry or an incidence geometry $X$ that is flag transitive, return `true` if and only if every flag of $X$ is contained in at least two chambers.

## `IsThin(X): CosetGeom -> BoolElt`

## `IsThin(X): IncGeom -> BoolElt`

Given either a coset geometry or an incidence geometry $X$ that is flag transitive, return `true` if and only if every flag of $X$ is contained in exactly two chambers.

## `IsThick(X): CosetGeom -> BoolElt`

## `IsThick(X): IncGeom -> BoolElt`

Given either a coset geometry or an incidence geometry $X$ that is flag transitive, return `true` if and only if every flag of the geometry is contained in exactly three chambers.

## `IsResiduallyConnected(X): IncGeom -> BoolElt`

## `IsRC(X): IncGeom -> BoolElt`

## `IsResiduallyConnected(X): CosetGeom -> BoolElt`

## `IsRC(X): CosetGeom -> BoolElt`

Given either a coset geometry or an incidence geometry $X$ that is flag transitive, return `true` if and only if every residue of rank at least two of $X$ has a connected incidence graph.

## `IsGraph(D): IncGeom -> GrphUnd`

Given an incidence geometry $D$, tests if this incidence geometry corresponds to a graph: $D$ must be of rank two and such that for one of the two types, say $e$, all elements of this type are incident with exactly two elements of the other type. Elements of type $e$ then correspond to edges of an undirected graph and elements of the other type to the vertices of that graph.

## `IsGraph(C): CosetGeom -> GrphUnd`

Given a coset geometry $C$, tests if this geometry corresponds to a graph: $C$ must be of rank two and one of the two maximal parabolic subgroups, say $G_e$, must contain the Borel subgroup as a subgroup of index 2. In that case, the cosets of $G_e$ correspond to edges of a graph and the cosets of the other maximal parabolic subgroup correspond to the vertices of this graph.