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Page 1: On inductively minimal geometries that satisfy the intersection property

Journal of Combinatorial Theory, Series A 111 (2005) 327–330

www.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

bDépartement de Mathématique, Université Libre de Bruxelles, C.P. 216 - Géométrie, Boulevard du Triomphe,B-1050 Bruxelles, Belgium

Received 11 October 2004

Communicated by Francis BuekenhoutAvailable online 12 February 2005

Abstract

We prove that, up to isomorphism, for a given positive integern, there is only one inductivelyminimal pair(�, Sym(n)) of rankn − 1 that satisfies 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. classified these inductively minimal pairs. In[3], Buekenhout and Cara provedseveral properties of these pairs. In[8], Cara studied truncations of these inductively minimalpairs. Finally, in[9], Cara et al. counted these inductively minimal pairs up to isomorphism.

In [10], Jacobs and Leemans described algorithms to test the intersection property on cosetgeometries. Using these algorithms, they checked the intersection property on inductivelyminimal geometries up ton = 6. These geometries are available for instance in[5]. They

E-mail addresses:[email protected](P. Cara),[email protected](D. Leemans).1Postdoctoral fellow of the Fund for Scientific Research-Flanders (Belgium).

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

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are the residually weakly primitive coset geometries of rankn with a connected diagramfor the symmetric groupsSym(n + 1) (see[5] for the definitions).

It turned out that for eachn�6, up to isomorphism, only one inductively minimal geom-etry satisfies the intersection property. It is the one with a linear diagram. In this paper, weprove that if(�, G) is an inductively minimal pair and� satisfies the intersection property,then� is unique up to isomorphism and has the following diagram.

The paper is organised as follows. In Section2, we recall some definitions and fix notation.In Section3, we prove the result announced in this introduction.

2. Definitions 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 whereX is the set of elements of�, ∗ is theincidence relation,t is the type function andI is the set of types of�. Given a typei ∈ I

and a flagF of �, we define thei-shadow�i (F ) as the set of elements of typei incidentwith F.

We define 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 flag F is eitherempty or equal to the i-shadow of a flag 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 flag F is eitherempty or equal to the i-shadow of a flag incident to x and F.

Let G be a group of automorphisms of� acting flag-transitively on�, that is,G actstransitively on the chambers of�.

As in [4] let (�, G) be calledminimal if | G | �(n + 1)! wheren =| I |. Let (�, G)

be calledinductively minimalif for any connected subsetJ of I and any flagF of �, witht (F ) = I\J , the pair(�F , GF ), whereGF is the group induced on the residue�F of theflagF in � by the stabilizer ofF, 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 withn in a fairly wild way as it is shown in[9].

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Note / Journal of Combinatorial Theory, Series A 111 (2005) 327–330 329

Theorem 3.1(Buekenhout et al.[4] ). Let I be a finite set ofn�1 elements. Let � be afinite, firm geometry over I with a connected digon diagram and let G be a flag-transitiveautomorphism group of�. Assume that(�, G) is an inductively minimal pair. Then:

1. � is thin and residually connected;2. G is isomorphic toSym(n+ 1) and for eachi ∈ I such that the residue of an element of

type i has a connected diagram, � hasn+1elements of type i on which G acts faithfully;3. The diagram of� has no minimal circuit of lengthl > 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 in3–5, there is, up to isomorphism, one and only one

inductively minimal pair(�, G) admitting this diagram.

Lemma 3.1. Let I be a finite set ofn�1 elements. Let� be a finite, firm geometry over Iwith a connected digon diagram and let G be a flag-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 Theorem3.1. �

Lemma 3.2. The inductivelyminimalgeometry�over thediagrammentioned inLemma3.1does not satisfy(IP).

Proof. This geometry is constructed in the following way. LetI = {1, 2, 3}. We take threecopiesX1, X2 andX3 of a setX of four points. The elements of� areX1 ∪ X2 ∪ X3. Thetype t (xi) = i for xi ∈ Xi . An elementxi ∈ Xi is incident to an elementxj ∈ Xj if andonly if they are distinct as elements ofX. Takex2 ∈ X2 andx3 ∈ X3 such thatx2 andx3are the same element ofX. Therefore,x2 andx3 are not incident. Since the 1-shadows ofx2andx3 are the elements ofX1 which are distinct fromx2 in X, we have�1(x2)∩�1(x3) �= ∅and we cannot find a flagF ′ incident to bothx2 andx3 such that�1(x2)∩�1(x3) = �1(F

′).Therefore,� is not (IP). �

Theorem 3.2. Let I be a finite set ofn�1 elements. Let� be a finite, firm geometry overI with a connected digon diagram and let G be a flag-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 Lemmas3.1and3.2, the only inductively minimal geometry which could satisfy(IP) is the one with a linear diagram. This geometry is constructed in the following way.

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Elements of typei (with i = 1, . . . , n) are thei-subsets of a setXof n+1 elements. Incidenceis symmetrized inclusion. By construction, this geometry obviously satisfies (IP). By point 6.of Theorem3.1, it is unique up to isomorphism.�

References

[1] F. Buekenhout, Diagrams for geometries and groups, J. Combin. Theory Ser. A 27 (1979) 121–151.[2] F. Buekenhout, Minimal flagtransitive geometries, in: A. Barlotti, A. Bichara, P.V. Ceccherini, G. Tallini

(Eds.), Combinatorics ’90 (Gaeta 1990), vol. 52, North-Holland, Amsterdam, 1992, pp. 69–70.[3] F. Buekenhout, P. Cara, Some properties of inductively minimal flag-transitive geometries, Bull. Belg. Math.

Soc. Simon Stevin 5 (1998) 213–219.[4] F. Buekenhout, P. Cara, M. Dehon, Inductively minimal flag-transitive geometries, in: N.L. Johnson (Ed.),

Mostly Finite Geometries, 1997, pp. 185–190.[5] F. Buekenhout, M. Dehon, D. Leemans, An atlas of residually weakly primitive geometries for small groups,

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

1 (1991) 45–78.[7] F. Buekenhout,A. Pasini, Finite diagram geometry extending buildings, in: Handbook of Incidence Geometry:

Buildings and Foundations, North-Holland, Amsterdam, 1995, pp. 1143–1254, (Chapter 22).[8] P. Cara, Truncations of inductively minimal geometries, Discrete Math. 267 (1–3) (2003) 63–74.[9] P. Cara, S. Lehman, D.V. Pasechnik, On the number of inductively minimal geometries, Theoret. Comput.

Sci. 263 (2001) 31–35.[10] P. Jacobs, D. Leemans, An algorithmic analysis of the intersection property, LMS J. Comput. Math. 7 (2004)

284–299.[11] A. Pasini, Diagram Geometries, Oxford University Press, Oxford, 1994.


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