The Flag-Transitive C 3-Geometries of Finite Order
Satoshi Yoshiara
Division of Mathematical Sciences Osaka Kyoiku University Kashiwara Osaka 582 Japan
DOI: 10.1023/A:1022480312547
Abstract
It is shown that a flag-transitive C 3-geometry of finite order ( x, y) with x 
2 is either a finite building of type C 3 (and hence the classical polar space for a 6-dimensional symplectic space, a 6-dimensional orthogonal space of plus type, a 6- or 7-dimensional hermitian space, a 7-dimensional orthogonal space, or an 8-dimensional orthogonal space of minus type) or the sporadic A7-geometry with 7 points.
2 is either a finite building of type C 3 (and hence the classical polar space for a 6-dimensional symplectic space, a 6-dimensional orthogonal space of plus type, a 6- or 7-dimensional hermitian space, a 7-dimensional orthogonal space, or an 8-dimensional orthogonal space of minus type) or the sporadic A7-geometry with 7 points.Pages: 251–284
Keywords: incidence geometry; C3-geometry; flag-transitivity; generalized quadrangle
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References
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16. M. Suzuki, "Group theory II," Grundlehren der mathematischen Wissenschaften 248, Springer, New York, 1986.
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19. H.N. Ward, "On Ree's series of simple groups," Trans. Amer. Math. Soc. 121 (1966), 62-89.
20. S. Yoshiara and A. Pasini, "On flag-transitive anomalous C3-geometries," Contr. Algebra and Geom. 34 (1993), 277-286.
2. F. Buekenhout, "The basic diagram of a geometry," in Geometries and Groups, Lecture Notes in Math. 893, Springer (1981), 1-29.
3. R.W. Carter, Simple Groups of Lie Type, John Wiley and Sons, London-New York-Sydney, 1989.
4. J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
5. W. Feit, "Finite projective planes and a question about primes," Proc. A.M.S. 108 (1990), 561-564.
6. B. Huppert, "Endlich gruppen I," Grundlehren der mathematischen Wissenschaften 134, Springer, Berlin- Heidelberg-New York, 1967.
7. W.M. Kantor, "Primitive permutation groups of odd degree, and an application to finite projective planes," J. Algebra 106 (1987), 15-45.
8. P.B. Kleidman, "The maximal subgroups of the Steinberg triality groups 3 D 4 ( q ) and of their automorphism groups," J. Algebra 115 (1988), 182-199.
9. G. Lunardon and A. Pasini, "A result on C3-geometries," Europ. J. Combin. 10 (1989), 265-271.
10. G. Lunardon and A. Pasini, "Finite Cn geometries: A survey," Note di Mathematica 10 (1990), 1-35.
11. A. Pasini, "On geometries of type C3 that are either buildings or flat," Bull. Soc. Math. Belg. B 38 (1986), 75-99.
12. A. Pasini, "On finite geometries of type C3 with thick lines," Note di Mathematica 6 (1986), 205-236.
13. A. Pasini, "Flag-transitive C3-geometries," Discrete Math. 117 (1993), 169-182.
14. A. Pasini, Diagram Geometries, Oxford UP., Oxford, 1994.
15. S.E. Payne and J.A. Thas, "Finite generalized quadrangles," Research Notes in Math. 110, Pitman, Boston, 1984.
16. M. Suzuki, "Group theory II," Grundlehren der mathematischen Wissenschaften 248, Springer, New York, 1986.
17. J.Tits, "Buildings of spherical type and finite BN-pairs," Lecture Notes in Math. 386, Springer, Berlin, 1974.
18. J. Tits, "Local approach to buildings," in Geometric Vein, Springer, New York, 1981, 519-547.
19. H.N. Ward, "On Ree's series of simple groups," Trans. Amer. Math. Soc. 121 (1966), 62-89.
20. S. Yoshiara and A. Pasini, "On flag-transitive anomalous C3-geometries," Contr. Algebra and Geom. 34 (1993), 277-286.