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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 Gx which is the stabilizer of x in G. The subgroups (Gx, x ∈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 IncidenceGeometry_Constructors (H142E8)

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


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

This means the conversion has been done successfully. So cg is the coset geome try corresponding to neumaier.


> 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 @}

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013