Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T12:38:19.606Z Has data issue: false hasContentIssue false
  • English
  • Français

An Algorithmic Analysis of the Intersection Property

Published online by Cambridge University Press:  01 February 2010

Pascale Jacobs  and
Dimitri Leemans
Pascale Jacobs
Affiliation:
Université Libre de Bruxelles, Département de Mathématique, CP. 216 – Géométrie, Boulevard du Triomphe, B-1050 Bruxelles, Belgium, the.jack@swing.be
Dimitri Leemans
Affiliation:
Université Libre de Bruxelles, Département de Mathématique, CP. 216 – Géométrie, Boulevard du Triomphe, B-1050 Bruxelles, Belgium, dleemans@ulb.ac.be, http://cso.ulb.ac.be/~dleemans/

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, efficient algorithms are given to test the intersection property and some of its variations on flag-transitive coset geometries. These algorithms are then applied to geometries of some sporadic groups, namely the Mathieu groups M11, M12, M22 and M23, the Janko groups J1, J2 and J3 and the Higman-Sims group HS.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 7 , 2004 , pp. 284 - 299
Copyright
Copyright © London Mathematical Society 2004

References

1. Biliotti, M. and Pasini, A., ‘Intersection properties in geometry’, Geom. Dedicata13(1982) 257–275.Google Scholar
2. Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system 1:the user language’, J. Symbolic Comput 3/4(1997)235–265.CrossRefGoogle Scholar
3. Buekenhout, F., ‘Diagrams for geometries and groups’, J. Combin. Theory Ser.A 27 (1979)121–151.CrossRefGoogle Scholar
4. Buekenhout, F., ‘On the geometry of diagrams’, Geom. Dedicata8(1979)253–257.Google Scholar
5. Buekenhout, F., ‘Separation and dimension in a graph’, Geom. Dedicata8(1979)297–298.Google Scholar
6. Buekenhout, F., ‘The basic diagram of a geometry’, Geometries and groups, Lecture Notes in Math.893 (ed. Aigner, M. and Jungnickel, D., Springer, Berlin, 1981)1–29.Google Scholar
7. Buekenhout, F., ‘The geometry of the finite simple groups’, Buildings and the geometry of diagrams, Lecture Notes in Math.1181 (ed.Rosati, L. A., Springer, Berlin, 1986)1–78.Google Scholar
8. Buekenhout, F., and Pasini, A.,‘Finite diagram geometry extending buildings’, Handbook of incidence geometry: buildings and foundations(North-Holland, Amsterdam, 1995)1143–1254.CrossRefGoogle Scholar
9. Buekenhout, F., Cara, P., Dehon, M., and Leemans, D., ‘Residually weakly primitive geometries of small almost simple groups: a synthesis’, Topics in diagram geometry,Quaderni Mat.12(ed.Pasini, A., Department of Mathematics, Seconda Universita di Napoli, Caserta, 2003) 1–27.Google Scholar
10. Cara, P. and Leemans, D., ‘On inductively minimal geometries that satisfy the intersection property’, preprint, Université Libre de Bruxelles, 2004, http://cso.ulb.ac.be/~dleemans/abstracts/imgip.html.Google Scholar
11. Dehon, M., ‘Classifying geometries with Cayley’, J. Symbolic Comput. 17 (1994) 259–276.CrossRefGoogle Scholar
12. Dehon, M. and Leemans, D., Supplement to: ‘Constructing coset geometries with MAGMA:an application to the sporadic groups M12 and J1’, 2001,http://cso.ulb.ac.be/~dleemans/abstracts/algo.html.Google Scholar
13. Dehon, M. and Leemans, D., ‘Constructing coset geometries with MAGMA: an application to the sporadic groups M12 and J1’, Atti Sem. Mat. Fis. Univ. Modena L (2002) 415–427.Google Scholar
14. Dehon, M. and Leemans, D. and Miller, X., ‘The residually weakly primitive and (IP)2geometries of M11’, preprint, Université Libre de Bruxelles, 1996, http://cso.ulb.ac.be/~dleemans/abstracts/mllrwpri.html.Google Scholar
15. Gottschalk, H. and Leemans, D., ‘The residually weakly primitive geometries of the Janko group J1’, Groups and geometries(ed.Pasini, A. et al. , Birkhauser, Basel, 1998)65–79.CrossRefGoogle Scholar
16. Leemans, D., ‘The residually weakly primitive geometries of the Suzuki simple group Sz(8)’, Groups St Andrews 1997, II, London Math. Soc. Lect. Note Ser. 261(ed.Campbell, C. M. et al. , Cambridge Univ. Press, Cambridge, 1999)517–526.CrossRefGoogle Scholar
17. Leemans, D., Supplement to:‘The residually weakly primitive geometries of J2’, 2001,http://cso.ulb.ac.be/~dleemans/abstracts/j2.html.Google Scholar
18. Leemans, D., Supplement to:‘The residually weakly primitive geometries of M22: The residually weakly primitive geometries of M22’, 2001, http://cso.ulb.ac.be/~dleemans/abstracts/m22.html.Google Scholar
19. Leemans, D., Supplement to:‘The residually weakly primitive geometries of J3’, 2003,http://cso.ulb.ac.be/~dleemans/abstracts/j3.html.Google Scholar
20. Leemans, D., Supplement to:‘The residually weakly primitive geometries of M23’,2003, http://cso.ulb.ac.be/~dleemans/abstracts/m23.html.Google Scholar
21. Leemans, D., Supplement to:‘The residually weakly primitive geometries of HS: The residually weakly primitive geometries of HS’,2003, http://cso.ulb.ac.be/~dleemans/abstracts/hs.html.Google Scholar
22. Pasini, A., Diagram geometries(Oxford Univ. Press, 1994).CrossRefGoogle Scholar
23. Tits, J., ‘Les groupes de Lie exceptionnels et leur interprétation géométrique’, Bull. Soc. Math. Belg.8(1956)48–81.Google Scholar
24. Tits, J., ‘Géométries polyédriques et groupes simples’, Atti 2a Riunione Groupem. Math. Express. Lat. Firenze (1962)66–88.Google Scholar
Supplementary material: File

JCM 7 Jacobs and Leemans Appendix A

Jacobs and Leemans Appendix A

Download JCM 7 Jacobs and Leemans Appendix A(File)
File 5.8 KB
Supplementary material: File

JCM 7 Jacobs and Leemans Appendix B

Jacobs and Leemans Appendix B

Download JCM 7 Jacobs and Leemans Appendix B(File)
File 6.7 MB