Abstract
Let S = (P, B, I) be a finite generalized quadrangle of order (s, t), s > 1, t > 1. Given a flag (p, L) of S, a (p, L)-collineation is a collineation θ of S which fixes each point on L and each line through p. For any line N incident with p, N ≠ L, and any point u incident with L, u ≠ p, the group G(p, L) of all (p, L)-collineations acts semiregularly on the lines M concurrent with N, p not incident with M, and on the points w collinear with u, w not incident with L. If the group G(p, L) is transitive on the lines M, or equivalently, on the points w, then we say that S is (p, L)-transitive. We prove that the finite generalized quadrangle S is (p, L)-transitive for all flags (p, L) if and only if S is classical or dual classical. Further, for any flag (p, L), we introduce the notion of (p, L)-desarguesian generalized quadrangle, a purely geometrical concept, and we prove that the finite generalized quadrangle S is (p, L)-desarguesian if and only if it is (p, L)-transitive.
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References
Fong, P. and Seitz, G.M. 1973/1974. Groups with a (B, N)-pair of rank 2, I and II. Invent. Math.,21:1–57, and 24:191–239.
Kiss, G. (1990). A generalization of Ostrom's theorem in generalized n-gons. Simon Stevin. 64: 309–317.
Payne, S.E. and Thas, J.A. 1984. Finite generalized quadrangles. London, Boston, Melbourne: Pitman.
Thas, J.A. and Van Maldeghem, H. 1991. Generalized Desargues configurations in generalized quadrangles. Bull. Soc. Math. Belg., 42: 713–722.
Thas, J.A., Payne, S.E. and Van Maldeghem, H. 1991. Half Moufang implies Moufang for finite generalized quadrangles, Invent. Math., 105: 153–156.
Tits, J. 1976. Classification of buildings of spherical type an Moufang polygons: a survey. In Teorie Combinatorie, Volume I (ed. B. Segre) Roma Accad. Naz. Lincei pp. 229–246.
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Communicated by R.L. Mullin
Research Associate of the National Fund for Scientific Research (Belgium).
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Van Maldeghem, H., Thas, J.A. & Payne, S. Desarguesian finite generalized quadrangles are classical or dual classical. Des Codes Crypt 1, 299–305 (1991). https://doi.org/10.1007/BF00124605
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DOI: https://doi.org/10.1007/BF00124605