Michel Dehon

Classifying Geometries with Cayley

In 1961, J.Tits has described a way to define a geometry from a group and a collection of subgroups. A lot of interesting geometrical objects arising from this definition have been studied by many geometers. The problem of finding all the geometries associated to a given group has been solved by hand, only for very small groups. A partial classification of the geometries of the Hall-Janko group has been recently obtained by M.Hermand, with the help of CAYLEY. Here we present a set of CAYLEY programmes to classify all the primitive, firm, residually connected and flag-transitive geometries associated to a given group G . As an application, we give the results obtained for the group W (E6).

On the existence of 2-designs S λ (2, 3, ∪) without repeated blocks

Abstract We prove that there exists an S λ (2, 3, v ) without repeated blocks if and only if λ ⩽ v −2, λυ ( υ −1)=0 (mod 6) and λ ( υ −1)=0 (mod 2).

All Geometries of the Mathieu Group Mll Based on Maximal Subgroups

Using a Cayley program, we get all firm, residually connected geometries whose rank-two residues satisfy the intersection. property, on which M ll acts flag-transitively, and in which the stabilizer of each element is a maximal subgroup of M ll.

On Flag-transitive Incidence Geometries of Rank 6 for the Mathieu Group M12

We show that the Mathieu group M12 does not have geometries of rank greater or equal to 6, satisfying the RWPRI and (IP)2 conditions. Our proof of this result is based on classifications of geometries of some subgroups of M12 which have been obtained using Magma programs.

Ranks of Incidence Matrices of t-designs Sλ(t, t+1, v)

Let S be an S λ ( t , t +1, v ) with v > t + 1 and let Rk p ( S ) be the rank over GF ( p ) of an incidence matrix of S ; we prove that R k p ( S ) = v if p ∤ t + 1 , and R k p ( S ) = v − 1 if p | t + 1 , except possibly when t = 2 and p = 2 or 3, or when t = 3 and p = 2.

Projective injections of geometries and their affine extensions

We present a general method allowing the construction geometries whose diagram is an extension of the diagram of a given geometry. Some applications of this construction process are described.

Planar Steiner triple systems

A regular planar Steiner triple system is a Steiner triple system provided with a family of non-trivial sub-systems of the same cardinality (called planes) such that (i) every set of 3 non collinear points is contained in exactly one plane and (ii) for every plane H and every disjoint block B, there are exactly α planes containing B and intersecting H in a block. We prove that a regular planar Steiner triple system is necessarily a projective space of dimension greater than 2 over GF(2), the 3-dimensional affine space over GF(3), an S(2, 3, 2 (6m+7) (3m2+3m+1)+1) with m⩾1, an S(2, 3, 171), an S(2, 3, 183) or an S(2, 3, 2055).

A geometry of rank 5 associated with PGO 5 (3)

We construct a thin, residually connected, primitive, and flag-transitive geometry of rank 5. Its residues of type {i, i + 1(mod 5)} (i = 0, …, 4) are hexagons; the other rank 2 residues are triangles.

ENANTIOMERIC LABELING OF REACTION GRAPHS

We examine systematically the possibility that a given reaction graph connects enantiomers. We define an enantiomeric labeling as a way to dispose enantiomers on the graph according to specified conditions. Since reaction graphs have to be symmetric, we have derived the set of enantiomeric labelings for every symmetric graph having less than 20 vertices. For some of these graphs, this set is empty; for others it contains only one labeling. Unexpectedly, it appears that some graphs have many such labelings which may even have different distances between enantiomers. As a result, the distance between enantiomers is not always the diameter of the graph.

The Geometry of Subspaces of an S(λ;2,3,v)

We define some linear spaces on the set of all proper subspaces of a triple system S(γ2,3,v). The connected components of these linear spaces are projective spaces of order 2 and punctured projective spaces of order 3, i.e. projective spaces of order 3 from which a point has been deleted. We show how these connected components can be used to find affine and projective factors in S(γ2,3,v).