# Conversion Functions

In this section we describe the functions that are available to convert incidence geometries and coset geometries in other objects.

## `IncidenceGeometry(C): CosetGeom -> IncGeom`

Construct the incidence geometry $IG$ from the coset geometry $C$. This is done using Tits’ algorithm described in the introduction of this chapter . The function returns one value: the incidence geometry $IG$.

## `CosetGeometry(D): IncGeom -> BoolElt, CosetGeom`

Convert the incidence geometry $D$ into a coset geometry.

If $D$ is an incidence geometry that can be converted into a coset geometry, the coset geometry isomorphic to it is constructed in the following way. The group $G$ of the coset geometry $CG$ is the automorphism group of $D$. Magma determines a chamber $C$ of $D$, that is a clique of the incidence graph of $D$ containing one element of each type. To every element $x$ in $C$, Magma associates a subgroup $G_x$ which is the stabilizer of $x$ in $G$. The subgroups $(G_x, x \in C)$ are the maximal parabolic subgroups of $CG$. In order to obtain a coset geometry combinatorially isomorphic to the incidence geometry we started with, the group $G$ must be transitive on every rank two truncation of $D$. If this condition is satisfied, the function returns a boolean set to the value `true` and the coset geometry $CG$. Otherwise, the function returns `false`.

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

If `IsGraph(D)` returns `true`, this function construct the undirected graph corresponding to the incidence geometry $D$.

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

If `IsGraph(C)` returns `true`, this function construct the undirected graph corresponding to the coset geometry $C$.

## `Example: Constructors (ex-cfd076)`

Taking back the last example for incidence geometries, we can convert the Neumaier geometry into a coset geometry by typing the following command (`neumaier` is the Neumaier geometry constructed above):

```magma
> ok,cg := CosetGeometry(neumaier);
> ok;
true

```

This means the conversion has been done successfully. So $cg$ is the coset geometry corresponding to `neumaier`.

```magma
> cg;
Coset geometry cg with 4 types
Group:
Permutation group acting on a set of cardinality 200
Order = 126000 = 2^4 * 3^2 * 5^3 * 7

Maximal Parabolic Subgroups:
Permutation group acting on a set of cardinality 200
Order = 2520 = 2^3 * 3^2 * 5 * 7
Permutation group acting on a set of cardinality 200
Order = 2520 = 2^3 * 3^2 * 5 * 7
Permutation group acting on a set of cardinality 200
Order = 2520 = 2^3 * 3^2 * 5 * 7
Permutation group acting on a set of cardinality 200
Order = 2520 = 2^3 * 3^2 * 5 * 7

Type Set:
{@ 1, 2, 3, 4 @}

```