# Elementary Invariants

## `Points(D): IncGeom -> SetIndx`

## `Elements(D): IncGeom -> SetIndx`

Given an incidence geometry $D$, return the set of elements of $D$. These elements are the points of the incidence graph of $D$.

## `Types(D): IncGeom -> SetIndx`

Given an incidence geometry $D$, return the set of types of $D$.

## `Types(C): CosetGeom -> SetIndx`

Given a coset geometry $C$, return the set of types of $C$.

## `Rank(D): IncGeom -> RngIntElt`

Given an incidence geometry $D$, return the rank of $D$, i.e. the cardinality of the set of types.

## `Rank(C): CosetGeom -> RngIntElt`

Given a coset geometry $C$, return the rank of $C$.

## `IncidenceGraph(D): IncGeom -> GrphUnd, GrphVertSet, GrphEdgeSet`

Given an incidence geometry $D$, return the incidence graph of $D$, its vertex set and its edge set.

We remark that this function is not implemented for coset geometries but we may convert a coset geometry into an incidence geometry using `IncidenceGeometry` and then compute its incidence graph.

## `Group(C): CosetGeom -> GrpPerm`

Given a coset geometry $C$, return the group from which $C$ is constructed.

## `MaxParabolics(C): CosetGeom -> SetIndx`

## `MaximalParabolics(C): CosetGeom -> SetIndx`

Given a coset geometry $C$, return an indexed set containing the maximal parabolics of $C$.

## `MinParabolics(C): CosetGeom -> SetIndx`

## `MinimalParabolics(C): CosetGeom -> SetIndx`

Given a coset geometry $C$, return an indexed set containing the minimal parabolics of $C$.

## `Borel(C): CosetGeom -> GrpPerm`

## `BorelSubgroup(C): CosetGeom -> GrpPerm`

Given a coset geometry $C$, return the *Borel subgroup* of $C$, i.e. the intersection of all maximal parabolic subgroups of $C$.

## `Kernel(C): CosetGeom -> GrpPerm`

Given a coset geometry $C$, return a permutation group which is its kernel, i.e. the subgroup of the Borel subgroup of $C$ that fixes all elements of the geometry $C$.

## `Kernels(C): CosetGeom -> SeqEnum`

Given a coset geometry $C$, return a sequence containing the $i$-kernel $K_i$ of each maximal parabolic subgroup $G_i$ of $C$. The $i$-kernel of the subgroup $G_i$ is the subgroup consisting of all the elements of $G_i$ that fix all the elements of the residue of $G_i$.

## `Quotient(C, K): CosetGeom, GrpPerm -> CosetGeom`

Given a coset geometry $C = (G; (G_i)_{i\in I})$ and a permutation group $K$, return the coset geometry $(G/K; (G_i/K)_{i\in I})$ provided that $K$ is a normal subgroup of $G$ and of all the maximal parabolic subgroups of $C$.