## An atlas of regular thin geometries for small groups

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## Abstract:

For some small groups, we give, up to isomorphism, an exhaustive list of all residually connected thin geometries on which these groups act regularly. We then show the utility of such an atlas by proving several results about smallest groups acting on a given diagram. The results have been obtained using a series of Magma programs.## References

- Michael Aschbacher,
*Flag structures on Tits geometries*, Geom. Dedicata**14**(1983), no. 1, 21–32. MR**701748**, DOI 10.1007/BF00182268 - Francis Buekenhout,
*Diagrams for geometries and groups*, J. Combin. Theory Ser. A**27**(1979), no. 2, 121–151. MR**542524**, DOI 10.1016/0097-3165(79)90041-4 - Francis Buekenhout,
*$(g,\,d^{\ast } ,\,d)$-gons*, Finite geometries (Pullman, Wash., 1981) Lecture Notes in Pure and Appl. Math., vol. 82, Dekker, New York, 1983, pp. 93–111. MR**690799** - F. Buekenhout (ed.),
*Handbook of incidence geometry*, North-Holland, Amsterdam, 1995. Buildings and foundations. MR**1360715** - F. Buekenhout and P. Cara,
*Some properties of inductively minimal flag-transitive geometries*, Bull. Belg. Math. Soc. 5 (1998), 213–219. - Walter Whiteley,
*Rigidity and polarity. I. Statics of sheet structures*, Geom. Dedicata**22**(1987), no. 3, 329–362. MR**887581**, DOI 10.1007/BF00147940 - —,
*Geometries of small almost simple groups based on maximal subgroups*, Bull. Belg. Math. Soc. - Simon Stevin Suppl. (1998). - Francis Buekenhout, Michel Dehon, and Dimitri Leemans,
*All geometries of the Mathieu group $M_{11}$ based on maximal subgroups*, Experiment. Math.**5**(1996), no. 2, 101–110. MR**1418957**, DOI 10.1080/10586458.1996.10504581 - —,
*An Atlas of residually weakly primitive geometries for small groups*, Mém. Acad. Royale Belg., Classe des Sciences (1996), To appear. - —,
*On flag-transitive incidence geometries of rank 6 for the Mathieu group M$_{12}$*, Groups and Geometries (A. Pasini et al., eds.), Birkhäuser, 1998, pp. 39–54. - Francis Buekenhout and Dimitri Leemans,
*On the list of finite primitive permutation groups of degree $\leq 50$*, J. Symbolic Comput.**22**(1996), no. 2, 215–225. MR**1422147**, DOI 10.1006/jsco.1996.0049 - —,
*On a geometry of Ivanov and Shpectorov for the O’Nan sporadic simple group*, J. Combin. Theory Ser. A 85 (1999), 148–164. - J. Cannon and W. Bosma,
*Handbook of magma functions*, Department of Pure Mathematics, University of Sydney, November 1994. - P. Cara,
*Truncations of inductively minimal geometries*, Preprint, 1997. - P. Cara, S. Lehman and D. Pasechnik,
*On the number of inductively minimal geometries*, Theoret. Comput. Sci. (to appear) - J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,
*$\Bbb {ATLAS}$ of finite groups*, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR**827219** - H. S. M. Coxeter and W. O. J. Moser,
*Generators and relations for discrete groups*, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 14, Springer-Verlag, Berlin-Göttingen-New York, 1965. MR**0174618** - Hans Cuypers,
*On a generalization of Fischer spaces*, Geom. Dedicata**34**(1990), no. 1, 67–87. MR**1058963**, DOI 10.1007/BF00150688 - M. Dehon and X. Miller,
*The residually weakly primitive and (IP)$_2$ geometries of M$_{11}$*, In preparation. - —,
*The residually weakly primitive and (IP)$_2$ geometries of U(4,2)*, In preparation. - M. Schönert et al.,
*GAP Version 3.4*, Lehrstuhl D für Mathematik, RWTH-Aachen, 1995. - H. Gottschalk,
*A classification of geometries associated with PSL(3,4)*, Diplomarbeit, Giessen, 1995. - H. Gottschalk and D. Leemans,
*The residually weakly primitive geometries of the Janko group J$_1$*, Groups and Geometries (A. Pasini et al., ed.), Birkhäuser, 1998, pp. 65–79. - D. Leemans,
*The residually weakly primitive geometries of the dihedral groups*, Atti Sem. Mat Fis. Univ. Modena (to appear). - —,
*The rank 2 geometries of the simple Suzuki groups Sz(q)*, Beiträge Algebra Geom.**39**(1998), no. 1, 97–120. - —,
*Thin geometries for the Suzuki simple group Sz(8)*, Proc. Third Int. Conf. on Finite Geometries and Combinatorics (F. De Clerck et al., ed.), vol. 5, Bull. Belg. Math. Soc. - Simon Stevin, 1998, pp. 373–387. - —,
*The residually weakly primitive geometries of the Suzuki simple group Sz(8)*, Proceedings of Groups St Andrews 1997 in Bath (C.M. Campbell et al., ed.), CUP, To appear. - P. McMullen and E. Schulte,
*Regular polytopes in ordinary space*, Discrete Comput. Geom.**17**(1997), no. 4, 449–478. Dedicated to Jörg M. Wills. MR**1455693**, DOI 10.1007/PL00009304 - J. Tits,
*Géométries polyédriques et groupes simples*, Atti 2a Riunione Groupem. Math. Express. Lat. Firenze (1962), 66–88.

## Additional Information

**Dimitri Leemans**- Affiliation: Université Libre de Bruxelles, Département de Mathématique, C.P.216- Géométrie, Boulevard du Triomphe, B-1050 Bruxelles
- MR Author ID: 613090
- ORCID: 0000-0002-4439-502X
- Email: dleemans@ulb.ac.be
- Received by editor(s): February 10, 1998
- Published electronically: May 17, 1999
- Additional Notes: This research was accomplished during a stay at the University of Sydney. We gratefully acknowledge support from the Fonds National de la Recherche Scientifique de Belgique and The University of Sydney.
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp.
**68**(1999), 1631-1647 - MSC (1991): Primary 51E24, 52B10, 20B99
- DOI: https://doi.org/10.1090/S0025-5718-99-01130-8
- MathSciNet review: 1654025