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C^ +-Groups

The concept of C^ +-groups has been introduced by Fernandes, Leemans and Weiss [MEFW] to generalise the study of rotational subgroups of Coxeter groups and study chiral geometries.

Let R be the set of generators of a group G. Suppose R:={α₁, ..., α_{r - 1}}. Define α₀:=1_{G^ +} and α_ij := α_i^{- 1}α_j for all 0≤i, j ≤r - 1. Let G_I := < α_ij | i, j ∈I > for I⊆{0, ..., r - 1}.

If the pair (G, R) satisfies the condition that forall I, J ⊆{0, ..., r - 1}, with |I|, |J| ≥2, G_I ∩G_J = G_I∩J, called the intersection property IP^ + (obtained in analogy with the intersection property of C-groups keeping only those equalities that involve subsets I and J of cardinality at least two), we say that (G, R) is a C^ +-group.

It follows immediately from the intersection property IP^ +, that R is an independent generating set for G, that means that α_i not∈< α_j : j != i >.

A group G whose set of generators satisfies the intersection property IP^ + is called a C^ +-group.

The B-diagram of G is a graph whose vertices are α₀ := Id(G) and α_i = G.i for each generator G.i of G. Two vertices are joined by an edge provided the order of α_i^{- 1}α_j is at least 3. Moreover, this edge is labelled with the order of α_i^{- 1}α_j.

The interest of C^ +-group is in that they permit also to construct chiral geometries. This is described in more details in [MEFW].

HasIntersectionPropertyPlus(G) : GrpPerm -> BoolElt

IsCPlusGroup(G) : GrpPerm -> BooElt

Given a permutation group G, determines whether or not it is a C^ +-group, that is whether or not its set of generators G.1, ..., G.n satisfy the intersection property IP^ +.

CosetGeometryFromCPlusGroup(G) : GrpPerm -> CosetGeom

Given a C^ +-group G, construct the coset geometry whose maximal parabolic subgroups are either generated by all but one generator of G, or the generators G.1^{- 1}G.i with i=2, ..., n where n is the number of generators of G. The constructed geometry is of rank n + 1.

BDiagram(G) : GrpPerm -> GrphUnd

Given a C^ +-group G, construct the B-diagram of G.

Example `IncidenceGeometry_cgrouptocosetgeometry (H151E15)`

C^ +-groups can be used to construct coset geometries as well. The user has to be careful that the corresponding geometry will have its rotation group acting on it. Let us again construct the 5-simplex using its rotation subgroup that is the alternating group A₆.

> Gp := sub<Alt(6)|(1,2,3),(1,2)(3,4),(1,2)(4,5),(1,2)(5,6)>;
> IsCPlusGroup(Gp);
true
> d,v,e := BDiagram(Gp);
> for x in e do print x,Label(x); end for;
{1, 2} 3
{2, 3} 3
{3, 4} 3
{4, 5} 3
> cg:=CosetGeometryFromCPlusGroup(Gp);
> IsThin(cg);
false

Above we see that the coset geometry constructed is not considered thin by Magma in terms of groups. This is because the stabilizer of a flag of rank 4 is the identity in A₆. To get the group S₆ acting, we have to convert the coset geometry into an incidence geometry and convert that incidence geometry in a coset geometry again. The latter one will have the automorphism group of the underlying incidence geometry acting on it, that is S₆.

> ig:=IncidenceGeometry(cg);
> ok,cg2:=CosetGeometry(ig);
> ok;
true
> IsThin(cg2);
true
> IsRC(cg2);
true
> IsFTGeometry(cg2);
true
> d,v,e:=Diagram(cg2);
> for x in v do print x,Label(x); end for;
1 [ 1, 15 ]
2 [ 1, 6 ]
3 [ 1, 15 ]
4 [ 1, 6 ]
5 [ 1, 20 ]
> for x in e do print x,Label(x); end for;
{1, 2} [ 3, 3, 3 ]
{1, 3} [ 2, 2, 2 ]
{1, 4} [ 2, 2, 2 ]
{1, 5} [ 3, 3, 3 ]
{2, 3} [ 2, 2, 2 ]
{2, 4} [ 2, 2, 2 ]
{2, 5} [ 2, 2, 2 ]
{3, 4} [ 3, 3, 3 ]
{3, 5} [ 3, 3, 3 ]
{4, 5} [ 2, 2, 2 ]