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Lax Embeddings of Generalized Quadrangles in Finite Projective Spaces
Author(s) -
Thas J. A.,
Van Maldeghem H.
Publication year - 2001
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611501012680
Subject(s) - mathematics , quadrangle , isomorphism (crystallography) , order (exchange) , combinatorics , class (philosophy) , dimension (graph theory) , space (punctuation) , projective space , generality , product (mathematics) , pure mathematics , discrete mathematics , projective test , geometry , computer science , artificial intelligence , psychology , finance , economics , chemistry , archaeology , crystal structure , psychotherapist , history , crystallography , operating system
A generalized quadrangle S is laxly embedded in a (finite) projective space PG( d , q ) if S is a subgeometry of the geometry of points and lines of PG( d , q ), with the only condition that the points of S generate the whole space PG( d , q ) (which one can always assume without loss of generality). In this paper, we classify thick laxly embedded quadrangles satisfying some additional hypotheses. The hypotheses are (a combination of) arestriction on the dimension d , a restriction on the parameters of S , and an assumption on the isomorphism class of S . In particular, the classification is complete in the following cases: (1) for d ⩾ 5; (2) for d =4 and S having ‘known’ order ( s , t ) with t ≠ s 2; (3) for d ⩾ 3 and S isomorphic to a finite Moufang quadrangle distinct from W ( s ) with s odd.As a by‐product, we obtain a new characterization theorem of the classical quadrangle H (4, s 2), and we also show that every generalized quadrangle of order ( s , s +2), with s >2, has at least one non‐regular line. 2000 Mathematics Subject Classification : 51E12.