An Atlas of residually weakly primitive geometries for small groups

Textes du quatrième de couverture du mémoire

Using a series of CAYLEY programs, we get all firm, residually connected geometries whose rank 2 residues satisfy the intersection property and on which a "small" flag-transitive group G is acting in such a way that for each flag F, the stabilizer G_F acts primitively on the elements of some type, in the residue of F.

The Atlas has two major parts. In the first one, we analyse 32 different groups. They are classified in three families : the alternating and symmetric groups up to degree 7, some projective groups (PSL₂(q) ≤ G ≤ Aut(PSL₂(q)) for q ≤ 8) and some groups of affine type (D_2p with p a prime number AGL₁(5), 7 : 3, AGL₁(7), AGL₁(8), AΓL₁(8), 3² : 4, 3² : D₈, 3² : 8, 3² : Q₈ ≡ M₉, AΓL₁(9), 3² : 2^.A₄, AGL₂(3), 11 : 5, AGL₁(11), 13 : 3, 13 : 4, 13 : 6 and AGL₁(13)). For each group we give a description of its structure, a subgroup pattern and the geometries satisfying the given properties. We also test some extra properties. We draw incidence and collinearity graphs for rank 2 geometries, incidence graphs for rank 3 geometries with a connected diagram and some incidence graphs for rank 4 geometries with a connected diagram.

In the second part, we list all geometries found in the first part. We classify them by rank, parameters and diagram connectivity.


Dr. Francis Buekenhout is born in 1937. He is Professor of Mathematics at the Université libre de Bruxelles. His fields of interest include buildings, incidence geometry, group theory, combinatorics and computer algebra. He is elected in 1992 at the Académie royale de Belgique.

Dr. Michel Dehon is born in 1952. He is Professor of Mathematics at the Université libre de Bruxelles. His fields of interest include incidence geometry, finite groups, designs, Steiner systems and computer algebra.

Dr. Dimitri Leemans is born in 1972. He is Assistant in Mathematics at the Université libre de Bruxelles. His fields of interest include incidence geometry, group theory, combinatorics and computer algebra.