Let us recall definitions of the properties described in this section.
An incidence geometry Γ is flag--transitive if for every two flags x, y of the same type of Γ, there exists an element g of Aut(Γ) such that g(x) = y. We also say that Aut(Γ) acts flag--transitively in this case.
Moreover, it is a flag--transitive geometry if it contains at least one chamber.
A coset geometry Γ(G; (Gi)i∈I) is flag--transitive if for every two flags x, y of the same type of Γ, there exists an element g of G such that g(x) = y. It is then a flag--transitive geometry since the set { (Gi)i∈I } is a chamber of Γ.
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.
Given a coset geometry C, return true if and only if the group of C acts flag--transitively on C.
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.
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.
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.
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.
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.
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 Ge, must contain the Borel subgroup as a subgroup of index 2. In that case, the cosets of Ge correspond to edges of a graph and the cosets of the other maximal parabolic subgroup correspond to the vertices of this graph.