Philippe Cara

Independent generating sets and geometries for symmetric groups

Abstract Julius Whiston showed that the size of an independent generating set in the symmetric group S n is at most n −1. We determine all sets meeting this bound. We also give some general remarks on the maximum size of an independent generating set of a group and its relationship to coset geometries for the group. In particular, we determine all coset geometries of maximum rank for the symmetric group S n for n >6.

Geometries of the Group PSL(2,11)

We determine all residually weakly primitive flag-transitive geometries for the groups PSL(2,11) and PGL(2,11). For the first of these we prove the existence by simple constructions while uniqueness, namely the fact that the lists are complete, relies on MAGMA programs. A central role is played by the subgroups Alt(5) in PSL(2,11). The highest rank of a geometry in our lists is four. Our work is related to various ‘atlases’ of coset geometries.

On the number of inductively minimal geometries

We count the number of inductively minimal geometries for any given rank by exhibiting a correspondence between the inductively minimal geometries of rank n and the trees with n + 1vertices. The proof of this correspondence uses the van Rooij-Wilf characterization of line graphs(see [11]).

The isomorphism problem for linear representations and their graphs The isomorphism problem for linear representations and their graphs

Abstract A linear representation T*n(K) of a point set K is a point-line geometry, embedded in a projective space PG(n+1; q), where K is contained in a hyperplane. We put constraints on K which ensure that every automorphism of T*n(K) is induced by a collineation of the ambient projective space. This allows us to show that, under certain conditions, two linear representations T*n(K) and T*n(K′) are isomorphic if and only if the point sets K and K′ are PΓL-equivalent. We also deal with the slightly more general problem of isomorphic incidence graphs of linear representations. In the last part of this paper, we give an explicit description of the group of automorphisms of T*n(K) that are induced by collineations of PG(n + 1; q).

Quotients of incidence geometries

We develop a theory for quotients of geometries and obtain sufficient conditions for the quotient of a geometry to be a geometry. These conditions are compared with earlier work on quotients, in particular by Pasini and Tits. We also explore geometric properties such as connectivity, firmness and transitivity conditions to determine when they are preserved under the quotienting operation. We show that the class of coset pregeometries, which contains all flag-transitive geometries, is closed under an appropriate quotienting operation.

An infinite family of Petersen geometries with nonlinear diagram

We construct new Petersen geometries. Starting with an inductively minimal geometry, we remove the elements of some well-chosen types and replace them with new objects. This yields an infinite family of Petersen geometries containing a family given bybuekenhout in [2]. We find a large amount of new Petersen geometries whose diagram is not linear.

On inductively minimal geometries that satisfy the intersection property

We prove that, up to isomorphism, for a given positive integer n, there is only one inductively minimal pair (Γ, Sym(n)) of rank n - 1 that satisfies the intersection property. Moreover, we show that the diagram of Γ is linear.

On the Number of Planes in Neumaier's A8-Geometry

In one of his papers [2], A. Neumaier constructed a rank 4 incidence geometry on which the alternating group of degree 8 acts flag-transitively. This geometry is quite important since its point residue is the famous A7-geometry which is known to be the only flag-transitive locally classical C3-geometry which is not a polar space (see [1]). By counting chambers, we prove that the A8-geometry has 70 planes. This can be found in a paper of Pasinis [4] without proof, but Neumaiers original paper only mentions 35 planes.

RWPRI Geometries for the Alternating Group A 8

We give an exhaustive list of the residually connected, flag-transitive geometries of the alternating group A 8 satisfying the (IP)2 and RWPRI properties. The list was obtained with the aid of a computer and the software package MAGMA[1]. The programs used are based on the algorithms described in [15] and [16]. Many geometries were predicted by a theory of inductively minimal geometries. For the remaining geometries, we give an explicit construction proving their existence. These constructions rely on various geometrical objects resulting from the well-known isomorphisms A 8 ≌ PSL(4, 2) ≌ PO; +(6, 2).