On Flat Flag-Transitive c.c *-Geometries
Barbara Baumeister
and Antonio Pasini
DOI: 10.1023/A:1008662500378
Abstract
We study flat flag-transitive c.c *-geometries. We prove that, apart from one exception related to Sym(6), all these geometries are gluings in the meaning of [6]. They are obtained by gluing two copies of an affine space over GF(2). There are several ways of gluing two copies of the n-dimensional affine space over GF(2). In one way, which deserves to be called the canonical one, we get a geometry with automorphism group G = 2 2 n ; L n(2) and covered by the truncated Coxeter complex of type D 2 n . The non-canonical ways give us geometries with smaller automorphism group ( G 
2 2 n ; (2 n-1) n) and which seldom (never ?) can be obtained as quotients of truncated Coxeter complexes.
2 2 n ; (2 n-1) n) and which seldom (never ?) can be obtained as quotients of truncated Coxeter complexes.Pages: 5–26
Keywords: diagram geometry; semi-biplane; amalgam of group
Full Text: PDF
References
1. M. Aschbacher, Finite Group Theory, Cambridge University Press, Cambridge, 1986.
2. B. Baumeister, Fahnentransitive Rang 3 Geometrien, Die Lokal Vollst\ddot andige Graphen Sind, Diplomarbeit, Frei Universit\ddot at Berlin, Summer
1992. P1: rbaP1: rba Journal of Algebraic Combinatorics KL365\?01(Baum) October 31, 1996 17:6 26 BAUMEISTER AND PASINI
3. B. Baumeister, “Two new sporadic semibiplanes related to M22,” European J. Comb. 15 (1994), 325-336.
4. B. Baumeister, On Flag-Transitive c.c* -Geometries, “Groups, Difference Sets and the Monster,” K.T. Arasu et al. (Eds.), de Gruyter, New York (1996), 3-21.
5. F. Buekenhout, “The geometry of the finite simple groups,” in Buildings and the Geometry of Diagrams, L.A. Rosati (Ed.), L.N. 1181, Springer (1986), 1-78.
6. F. Buekenhout, C. Huybrechts, and A. Pasini, “Parallelism in diagram geometry,” Bulletin Belgian Math. Soc. Simon Stevin 1 (1994), 355-397.
7. P. Cameron, “On groups of degree n and n - 1, and highly transitive symmetric edge colourings,” J. London Math. Soc. (2) 9 (1975), 385-391.
8. P. Cameron, “Finite permutation groups and finite simple groups,” Bull. London Math. Soc. 13 (1981), 1-22.
9. P. Cameron and G. Korchmaros, “One-factorizations of complete graphs with a doubly transitive automorphism group,” Bull. London Math. Soc. 25 (1993), 1-6.
10. J. Conway, R. Curtis, S. Norton, R. Parker, and R. Wilson, Atlas of Finite Groups, Oxford Uuniversity Press, 1985.
11. A. Del Fra, private communication.
12. D. Hughes, “Biplanes and semibiplanes,” in Combinatorics, L.N. 686, Springer (1978), 55-58.
13. B. Huppert, Endlichen Gruppen I, Springer, Berlin, 1979.
14. Z. Janko and T. Van Trung, “Two new semibiplanes,” J. Comb. Th. A 33 (1982), 102-105.
15. P. Kleidmann, “The low-dimensional finite simple classical groups and their subgroups,” Longman Research Notes in Mathematics, to appear.
16. D. K\ddot onig, Theorie der Endlichen und Unendlichen Graphen, Teubner, 1936.
17. G. Korchmaros, “Cyclic one-factorizations with an invariant one-factor of the complete graph,” Ars Combinatoria 27 (1989), 133-138.
18. C. Levefre-Percsy and L. van Nypelseer, “Finite rank 3 geometries with affine planes and dual affine point residues,” Discrete Math. 84 (1990), 161-167.
19. M. Liebeck, “The affine permutation groups of rank 3,” Proc. London Math. Soc. (3) 54 (1987), 477-516.
20. A. Pasini, “Gluing two affine spaces,” Bull. Soc. Math. Belgique-Simon Stevin 3 (1996), 25-40.
21. A. Pasini, Diagram Geometries, Oxford University Press, Oxford, 1994.
22. B. Baumeister and D. Pasechnik, “The Universal Cover of Certain Semibiplanes,” preprint.
2. B. Baumeister, Fahnentransitive Rang 3 Geometrien, Die Lokal Vollst\ddot andige Graphen Sind, Diplomarbeit, Frei Universit\ddot at Berlin, Summer
1992. P1: rbaP1: rba Journal of Algebraic Combinatorics KL365\?01(Baum) October 31, 1996 17:6 26 BAUMEISTER AND PASINI
3. B. Baumeister, “Two new sporadic semibiplanes related to M22,” European J. Comb. 15 (1994), 325-336.
4. B. Baumeister, On Flag-Transitive c.c* -Geometries, “Groups, Difference Sets and the Monster,” K.T. Arasu et al. (Eds.), de Gruyter, New York (1996), 3-21.
5. F. Buekenhout, “The geometry of the finite simple groups,” in Buildings and the Geometry of Diagrams, L.A. Rosati (Ed.), L.N. 1181, Springer (1986), 1-78.
6. F. Buekenhout, C. Huybrechts, and A. Pasini, “Parallelism in diagram geometry,” Bulletin Belgian Math. Soc. Simon Stevin 1 (1994), 355-397.
7. P. Cameron, “On groups of degree n and n - 1, and highly transitive symmetric edge colourings,” J. London Math. Soc. (2) 9 (1975), 385-391.
8. P. Cameron, “Finite permutation groups and finite simple groups,” Bull. London Math. Soc. 13 (1981), 1-22.
9. P. Cameron and G. Korchmaros, “One-factorizations of complete graphs with a doubly transitive automorphism group,” Bull. London Math. Soc. 25 (1993), 1-6.
10. J. Conway, R. Curtis, S. Norton, R. Parker, and R. Wilson, Atlas of Finite Groups, Oxford Uuniversity Press, 1985.
11. A. Del Fra, private communication.
12. D. Hughes, “Biplanes and semibiplanes,” in Combinatorics, L.N. 686, Springer (1978), 55-58.
13. B. Huppert, Endlichen Gruppen I, Springer, Berlin, 1979.
14. Z. Janko and T. Van Trung, “Two new semibiplanes,” J. Comb. Th. A 33 (1982), 102-105.
15. P. Kleidmann, “The low-dimensional finite simple classical groups and their subgroups,” Longman Research Notes in Mathematics, to appear.
16. D. K\ddot onig, Theorie der Endlichen und Unendlichen Graphen, Teubner, 1936.
17. G. Korchmaros, “Cyclic one-factorizations with an invariant one-factor of the complete graph,” Ars Combinatoria 27 (1989), 133-138.
18. C. Levefre-Percsy and L. van Nypelseer, “Finite rank 3 geometries with affine planes and dual affine point residues,” Discrete Math. 84 (1990), 161-167.
19. M. Liebeck, “The affine permutation groups of rank 3,” Proc. London Math. Soc. (3) 54 (1987), 477-516.
20. A. Pasini, “Gluing two affine spaces,” Bull. Soc. Math. Belgique-Simon Stevin 3 (1996), 25-40.
21. A. Pasini, Diagram Geometries, Oxford University Press, Oxford, 1994.
22. B. Baumeister and D. Pasechnik, “The Universal Cover of Certain Semibiplanes,” preprint.