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Journal of Combinatorial Theory, Series A 111 (2005) 327330www.elsevier.com/locate/jcta

Note

On inductively minimal geometries that satisfy theintersection property

Philippe Caraa,1, Dimitri LeemansbaDepartment of Mathematics, Vrije Universiteit Brussels, Pleinlaan 2, B-1050 Brussel, Belgium

bDpartement de Mathmatique, Universit Libre de Bruxelles, C.P. 216 - Gomtrie, Boulevard du Triomphe,B-1050 Bruxelles, Belgium

Received 11 October 2004

Communicated by Francis BuekenhoutAvailable online 12 February 2005

AbstractWe prove that, up to isomorphism, for a given positive integer n, there is only one inductively

minimal pair (, Sym(n)) of rank n 1 that satises the intersection property. Moreover, we showthat the diagram of is linear. 2005 Elsevier Inc. All rights reserved.MSC: 51E24

Keywords: Incidence geometry; Inductively minimal; Intersection property

1. Introduction

Inductively minimal pairs (,G) were introduced by Buekenhout in [2]. In [4], Bueken-hout et al. classied these inductively minimal pairs. In [3], Buekenhout and Cara provedseveral properties of these pairs. In [8], Cara studied truncations of these inductivelyminimalpairs. Finally, in [9], Cara et al. counted these inductively minimal pairs up to isomorphism.In [10], Jacobs andLeemans described algorithms to test the intersection property on coset

geometries. Using these algorithms, they checked the intersection property on inductivelyminimal geometries up to n = 6. These geometries are available for instance in [5]. They

E-mail addresses: pcara@vub.ac.be (P. Cara), dleemans@ulb.ac.be (D. Leemans).1Postdoctoral fellow of the Fund for Scientic Research-Flanders (Belgium).

0097-3165/$ - see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jcta.2004.12.006

328 Note / Journal of Combinatorial Theory, Series A 111 (2005) 327330

are the residually weakly primitive coset geometries of rank n with a connected diagramfor the symmetric groups Sym(n+ 1) (see [5] for the denitions).It turned out that for each n6, up to isomorphism, only one inductively minimal geom-

etry satises the intersection property. It is the one with a linear diagram. In this paper, weprove that if (,G) is an inductively minimal pair and satises the intersection property,then is unique up to isomorphism and has the following diagram.

The paper is organised as follows. In Section 2, we recall some denitions and x notation.In Section 3, we prove the result announced in this introduction.

2. Denitions and notation

We assume knowledge of the basic notions in incidence geometry as they are given forinstance in [7] or [11].Let (X, , t, I ) be an incidence geometry where X is the set of elements of , is the

incidence relation, t is the type function and I is the set of types of . Given a type i Iand a ag F of , we dene the i-shadow i (F ) as the set of elements of type i incidentwith F.We dene the intersection property (IP) as it appears in [1].

(IP) For every type i, the intersection of the i-shadows of an element x and a ag F is eitherempty or equal to the i-shadow of a ag incident to x and F. The same holds on theresidues.

As mentioned in [6], this condition is equivalent to the following one.(IP) For every type i, the intersection of the i-shadows of an element x and a ag F is either

empty or equal to the i-shadow of a ag incident to x and F.Let G be a group of automorphisms of acting ag-transitively on , that is, G acts

transitively on the chambers of .As in [4] let (,G) be called minimal if | G | (n + 1)! where n =| I |. Let (,G)

be called inductively minimal if for any connected subset J of I and any ag F of , witht (F ) = I\J , the pair (F ,GF ), where GF is the group induced on the residue F of theag F in by the stabilizer of F, is minimal.

3. Inductively minimal geometries and the intersection property

Buekenhout et al. show in [4] that a full control can be achieved on inductively minimalpairs although their number grows with n in a fairly wild way as it is shown in [9].

Note / Journal of Combinatorial Theory, Series A 111 (2005) 327330 329

Theorem 3.1 (Buekenhout et al. [4]). Let I be a nite set of n1 elements. Let be anite, rm geometry over I with a connected digon diagram and let G be a ag-transitiveautomorphism group of . Assume that (,G) is an inductively minimal pair. Then:1. is thin and residually connected;2. G is isomorphic to Sym(n+ 1) and for each i I such that the residue of an element of

type i has a connected diagram, has n+1 elements of type i on which G acts faithfully;3. The diagram of has no minimal circuit of length l > 3;4. every edge of the diagram is on a unique maximal clique;5. each vertex of the diagram is either on one or two maximal cliques of the diagram;6. for any connected diagram as in 35, there is, up to isomorphism, one and only one

inductively minimal pair (,G) admitting this diagram.

Lemma 3.1. Let I be a nite set of n1 elements. Let be a nite, rm geometry over Iwith a connected digon diagram and let G be a ag-transitive automorphism group of .Assume that (,G) is an inductively minimal pair. If the diagram of is nonlinear, then has a residue of rank three over the following diagram.

Proof. This is a direct consequence of 3 and 5 of Theorem 3.1.

Lemma 3.2. The inductivelyminimal geometryover thediagrammentioned inLemma3.1does not satisfy (IP).

Proof. This geometry is constructed in the following way. Let I = {1, 2, 3}. We take threecopies X1, X2 and X3 of a set X of four points. The elements of are X1 X2 X3. Thetype t (xi) = i for xi Xi . An element xi Xi is incident to an element xj Xj if andonly if they are distinct as elements of X. Take x2 X2 and x3 X3 such that x2 and x3are the same element of X. Therefore, x2 and x3 are not incident. Since the 1-shadows of x2and x3 are the elements ofX1 which are distinct from x2 in X, we have 1(x2)1(x3) = and we cannot nd a ag F incident to both x2 and x3 such that 1(x2)1(x3) = 1(F ).Therefore, is not (IP).

Theorem 3.2. Let I be a nite set of n1 elements. Let be a nite, rm geometry overI with a connected digon diagram and let G be a ag-transitive automorphism group of .Assume that (,G) is an inductively minimal pair satisfying (IP). Then is unique up toisomorphism. Moreover, it has a linear diagram.

Proof. By Lemmas 3.1 and 3.2, the only inductively minimal geometry which could satisfy(IP) is the one with a linear diagram. This geometry is constructed in the following way.

330 Note / Journal of Combinatorial Theory, Series A 111 (2005) 327330

Elements of type i (with i = 1, . . . , n) are the i-subsets of a setX ofn+1 elements. Incidenceis symmetrized inclusion.By construction, this geometry obviously satises (IP). Bypoint 6.of Theorem 3.1, it is unique up to isomorphism.

References

[1] F. Buekenhout, Diagrams for geometries and groups, J. Combin. Theory Ser. A 27 (1979) 121151.[2] F. Buekenhout, Minimal agtransitive geometries, in: A. Barlotti, A. Bichara, P.V. Ceccherini, G. Tallini

(Eds.), Combinatorics 90 (Gaeta 1990), vol. 52, North-Holland, Amsterdam, 1992, pp. 6970.[3] F. Buekenhout, P. Cara, Some properties of inductively minimal ag-transitive geometries, Bull. Belg. Math.

Soc. Simon Stevin 5 (1998) 213219.[4] F. Buekenhout, P. Cara, M. Dehon, Inductively minimal ag-transitive geometries, in: N.L. Johnson (Ed.),

Mostly Finite Geometries, 1997, pp. 185190.[5] F. Buekenhout, M. Dehon, D. Leemans, An atlas of residually weakly primitive geometries for small groups,

Mm. Cl. Sci., Coll. 8, Ser. 3, Tome XIV. Acad. Roy. Belgique (1999).[6] F. Buekenhout, M. Hermand, On ag-transitive geometries and groups, Travaux Math. Univ. Libre Bruxelles

1 (1991) 4578.[7] F. Buekenhout,A. Pasini, Finite diagramgeometry extending buildings, in: Handbook of IncidenceGeometry:

Buildings and Foundations, North-Holland, Amsterdam, 1995, pp. 11431254, (Chapter 22).[8] P. Cara, Truncations of inductively minimal geometries, Discrete Math. 267 (13) (2003) 6374.[9] P. Cara, S. Lehman, D.V. Pasechnik, On the number of inductively minimal geometries, Theoret. Comput.

Sci. 263 (2001) 3135.[10] P. Jacobs, D. Leemans, An algorithmic analysis of the intersection property, LMS J. Comput. Math. 7 (2004)

284299.[11] A. Pasini, Diagram Geometries, Oxford University Press, Oxford, 1994.

On inductively minimal geometries that satisfy the intersection propertyIntroductionDefinitions and notationInductively minimal geometries and the intersection propertyReferences