Alice Devillers Homepage

Who am I?
My coordinates
My PhD thesis
My bibliography

Who am I ?

I was born on the 22nd of September 1976. I grew up in Brussels, Belgium. From 1994 to 1998, I studied mathematics at the Université Libre de Bruxelles.
I was (October 1998-January 2002) a PhD student in the Service de Géométrie of the Département de Mathématiques ( Université Libre de Bruxelles ). I had a grant from the Fonds National de la Recherche Scientifique. I defended my PhD on January 24, 2002. From October 2002 to September 2003, I worked at Michigan State University with Professor Jonathan Hall, thanks to the Belgian American Education Foundation (BAEF). Then I was back in Brussels until 2008, as a Chargé de Recherche of the FNRS. Since September 2008, I am a Research Associate at The University of Western Australia (Perth), working in Cheryl Praeger's team. I am a member of the Centre for the Mathematics of Symmetry and Computation. Since August 2010, I have been promoted to to Reseacrh Assistant Professor. I am also Collaborateur Scientifique at ULB.

My coordinates

Devillers Alice

The University of Western Australia
School of Mathematics and Statistics
35 Stirling Highway
Crawley 6009
Western Australia
Emails: adevil@ulb.ac.be, adevil@maths.uwa.edu.au
Phone: +61 8 6488 3228
Fax: +61 8 6488 1028

My PhD thesis

My PH-D Thesis ( PDF (1 mb) | PS (600 kb) )
Errata and misprints
Revised version of my PH-D Thesis PDF or PS

My bibliography

Devillers A. and Doyen J., "Homogeneous and ultrahomogeneous linear spaces", Journal of Combinatorial Theory Ser. A 84, pp. 236-241, 1998.
Devillers A. , "d-homogeneous and d-ultrahomogeneous linear spaces", Journal of Combinatorial Designs, 8(5), pp. 321-329, 2000.
Abstract: A linear space S is d-homogeneous if, whenever the linear structures induced on two subsets S₁ and S₂ of cardinality at most d are isomorphic, there is at least one automorphism of S mapping S₁ onto S₂. S is called d-ultrahomogeneous if each isomorphism between the linear structures induced on two subsets of cardinality at most d can be extended into an automorphism of S. We have proved (without any finiteness assumption) that every 6-homogeneous linear space is homogeneous (that is d-homogeneous for every positive integer d). Here we classify completely the finite non-trivial linear spaces that are d-homogeneous for d≥ 4 or d-ultrahomogeneous for d≥ 3. We also prove an existence theorem for infinite non-trivial 4-ultrahomogeneous linear spaces.
Devillers A. , "Ultrahomogeneous semilinear spaces", Proceedings of the London Mathematical Society, 84, pp.35-58, 2002. Abstract:A semilinear space S is ultrahomogeneous if each isomorphism between the semilinear structures induced on two finite subsets can be extended to an automorphism of S. We give a complete classification of all finite ultrahomogeneous semilinear spaces. Our theorem extendss a result of Gardiner on graphs and a result of Devillers and Doyen on linear spaces.
Devillers A. , "A classification of finite homogeneous semilinear spaces", Advances in Geometry 2 (4), 307-328, 2002.
Abstract: A semilinear space S is homogeneous if, whenever the semilinear structures induced on two finite subsets S₁ and S₂ of S are isomorphic, there is at least one automorphism of S mapping S₁ onto S₂. We give a complete classification of all finite homogeneous semilinear spaces. Our theorem extends a result of Ronse on graphs and a result of Devillers and Doyen on linear spaces.
Devillers A. , "Homogeneous and ultrahomogeneous Steiner systems", Journal of Combinatorial designs 11 (3), 153-161, 2003.
Abstract: A Steiner system (or t-(v,k,1) design) S is said to be homogeneous if, whenever the substructures induced on two finite subsets S₁ and S₂ of S are isomorphic, there is at least one automorphism of S mapping S₁ onto S₂, and is said to be ultrahomogeneous if each isomorphism between the substructures induced on two finite subsets of S can be extended to an automorphism of S. We give a complete classification of all homogeneous and ultrahomogeneous Steiner systems.
Devillers A. , "A classification of finite partial linear spaces with a primitive rank 3 automorphism group of almost simple type", Innovations in Incidence Geometry, 2, pp. 131-177, 2005.
Abstract: A partial linear space is a non-empty set of points, provided with a collection of subsets called lines such that any pair of points is contained in at most one line and every line contains at least two points. Graphs and linear spaces are particular cases of partial linear spaces. A partial linear space which is not a graph or a linear space is called proper. A partial linear space is said to be 2-ultrahomogeneous if it admits an automorphism group which is transitive on the ordered pairs of collinear points and on the ordered pairs of non-collinear points. For proper partial linear spaces, this means that the automorphism group is of rank 3. In this paper, we give a complete classification of all finite 2-ultrahomogeneous proper partial linear spaces admitting a primitive rank 3 automorphism group of almost simple type.
Devillers A. and J.I. Hall, "Rank 3 Latin square designs", Journal of Combinatorial Theory (A), 113, pp. 894-902 (2006).
Abstract: A Latin square design whose automorphism group is transitive of rank at most 3 on points must come from the multiplication table of an elementary abelian p-group, for some prime p.
Devillers A. and Hendrik Van Maldeghem, "Partial linear spaces built on hexagons" , European Journal of Combinatorics, 28(3), pp. 901-915 (2007).
Abstract: We define four families of geometries with as point graph the graph (or its dual) of all elliptic hyperplanes of a given parabolic quadric in any finite 6-dimensional projective space, where adjacency is given by intersecting in a tangent 4-space. One of the classes consist of (known) semi-partial geometries. Our main result determines the complete automorphism groups of all these geometries.
Devillers A. and Bernhard Muhlherr, "On the simple connectedness of certain subsets of buildings" , Forum Mathematicum, 19(6), pp. 955-970 (2007).
Abstract: We prove a rank 3 criterion for the simple connectedness of certain subsets of buildings and we give two applications of this criterion. The first generalizes a result of Tits for Chevalley groups to 3-spherical Kac-Moody groups. The second is the proof of the simple connectedness of certain flipflop geometries introduced in a paper by Bennet-Gramlich-Hoffmann-Shpectorov.
Devillers A., "A classification of finite partial linear spaces with a primitive rank 3 automorphism group of grid type" , European Journal of Combinatorics, 29(1), pp. 268-272 (2008).
Abstract: A partial linear space is a non-empty set of $points$, provided with a collection of subsets called $lines$ such that any pair of points is contained in at most one line and every line contains at least two points. Graphs and linear spaces are particular cases of partial linear spaces. A partial linear space which is neither a graph nor a linear space is called proper. The aim of this paper is to classify the finite proper partial linear spaces with a primitive rank 3 automorphism group of grid type.
Alice Devillers, Michael Giudici, Cai Heng Li and Cheryl E. Praeger, "Primitive decompositions of Johnson graphs" , Journal of Combinatorial Theory (A) 115 (6), 925-966 (2008).
Abstract: A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts.
Alice Devillers, Michael Giudici, "Involution graphs where the product of two adjacent vertices has order three" , J. Aust. Math. Soc. 85, 305-322 (2008).
Abstract: An S₃-involution graph for a group G is a graph with vertex set a union of conjugacy classes of involutions of G such that two involutions are adjacent if they generate an S₃-subgroup in a particular set of conjugacy classes. We investigate such graphs in general and also for the case where G = PSL(2, q).
Alice Devillers, Ralf Gramlich, Bernhard Muhlherr, "The sphericity of the complex of non-degenerate subspaces" , Journal of the London Mathematical Society 79 (3), 684-700 (2009).
Abstract: We prove that the complex of proper non-trivial non-degenerate subspaces of a finitedimensional vector space endowed with a non-degenerate sesquilinear form is homotopy equivalent to a wedge of spheres. Additionally, we show that the same is true for a slightly wider class of simplicial complexes, the so-called generalized Phan geometries of type An. These generalized Phan geometries occur as relative links of the filtration studied previously by the third and last author, whose sphericity implies topological finiteness properties of suitable arithmetic groups and allows for a revision of Phan's group-theoretical local recognition of suitable finite groups of Lie type with simply laced diagrams.
Alice Devillers, Michael Giudici, Cai Heng Li and Cheryl E. Praeger, "Some graphs related to the small Mathieu groups" , European J. Combin., 31 335-348 (2010).
Abstract: Two different constructions are given of a rank 8 arc-transitive graph with 165 vertices and valency 8, whose automorphism group is M₁₁. One involves 3-subsets of an 11-set while the other involves 4-subsets of a 12-set, and the constructions are linked with the Witt designs on 11, 12 and 24 points. Four different constructions are given of a rank 9 arc-transitive graph with 55 vertices and valency 6 whose automorphism group is PSL(2,11). This graph occurs as a subgraph of the M₁₁ graph, and two of the constructions involve 2-subsets of an 11-set while the remaining two involve 3-subsets of an 11-set. The PSL(2,11) and M₁₁ graphs occur as the second and third members of a tower of graphs defined on a conjugacy class of involutions of the simple groups A₅, PSL(2,11), M₁₁ and M₁₂ with two involutions adjacent if they generate a special S₃. The first graph in the tower is the line graph of the Petersen graph while the fourth is the Johnson graph J(12,4). The graphs also arise as collineation graphs of rank two truncations of various residues of certain P-geometries.
Alice Devillers, Bernhard Muhlherr, Hendrik Van Maldeghem, "Codistances of 3-spherical buildings" , submitted.
Abstract: We show that a 3-spherical building in which each rank 2 residue is connected far away from a chamber, and each rank 3 residue is simply 2-connected far away from a chamber, admits a twinning (i.e., is one half of a twin building) as soon as it admits a codistance, i.e., a twinning with a single chamber.
Alice Devillers, Michael Giudici, Cai Heng Li and Cheryl E. Praeger, "Locally s-distance transitive graphs", , submitted.
Abstract: We give a unified approach to analysing, for each positive integer s, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s-arc transitive graphs. A graph is in the class if it is connected and if, for each vertex v, the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v, for each i from 1 to s. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for s>1, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph, or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex orbits, or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups.
Alice Devillers, Michael Giudici, Cai Heng Li and Cheryl E. Praeger, "An infinite family of biquasiprimitive 2-arc transitive cubic graphs", submitted.
Abstract: A new infinite family of bipartite cubic 3-arc transitive graphs is constructed and studied. They provide the first known examples admitting a 2-arc transitive vertex-biquasiprimitive group of automorphisms for which the stabiliser of the biparts is not quasiprimitive on either bipart.
Ashraf Daneshkhah, Alice Devillers and Cheryl E. Praeger, "Symmetry properties of subdivision graphs", submitted.
Abstract: The subdivision graph S(&Sigma) of a graph &Sigma is obtained from &Sigma by `adding a vertex' in the middle of every edge. Various symmetry properties of S(&Sigma) are studied. We prove that, for a connected graph &Sigma, S(&Sigma) is locally s-arc transitive if and only if &Sigma is ⌊(s+1)/2⌋-arc transitive. The diameter of S(&Sigma) is 2d+&delta, where &Sigma has diameter d and 0≤ &delta≤ 2, and local s-distance transitivity of S(&Sigma) is defined for 1≤ s≤ 2d+&delta. In the general case where s≤ 2d-1 we prove that S(&Sigma) is locally s-distance transitive if and only if &Sigma is ⌊(s+1)/2⌋-arc transitive. For the remaining values of s, namely 2d≤ s≤ 2d+&delta, we classify the graphs &Sigma for which S(&Sigma) is locally s-distance transitive in the cases, s≤ 5 and s≥ 15+&delta. The cases max{2d, 6}≤ s≤ min{2d+&delta, 14+&delta} remain open.
Alice Devillers, Michael Giudici, Cai Heng Li, Geoffrey Pearce and Cheryl E.Praeger, "On imprimitive rank 3 permutation groups", submitted.
Abstract: A classification is given of rank 3 group actions which are quasiprimitive but not primitive. There are two infinite families and a finite number of individual imprimitive examples. When combined with earlier work of Bannai, Kantor, Liebler, Liebeck and Saxl, this yields a classification of all quasiprimitive rank 3 permutation groups. Our classification is achieved by first classifying imprimitive almost simple permutation groups which induce a 2-transitive action on a block system and for which a block stabiliser acts 2-transitively on the block. We also determine those imprimitive rank 3 permutation groups G such that the induced action on a block is almost simple and G does not contain the full socle of the natural wreath product in which G embeds.