Characterizations of generalized quadrangles by generalized homologies |
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Authors: | J A Thas |
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Institution: | Seminar of Geometry and Combinatorics, State University of Ghent, Krijgslaan 281, 9000 Ghent, Belgium |
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Abstract: | Let S = (P, B, I) be a generalized quadrangle of order (s, t). For x, y P, x y, let
(x, y) be the group of all collineations of S fixing x and y linewise. If z {x, y}, then the set of all points incident with the line xz (resp. yz) is denoted by
(resp.
). The generalized quadrangle S = (P, B, I) is said to be (x, y)-transitive, x y, if
(x, y) is transitive on each set
and
. If S = (P, B, I) is a generalized quadrangle of order (s, t), s > 1 and t > 1, which is (x, y)-transitive for all x, y P with x y, then it is proved that we have one of the following: (i) S W(s), (ii) S Q(4, s), (iii) S H(4, s), (iv) S Q(5, s), (v) s = t2 and all points are regular. |
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Keywords: | |
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