### A Characterization ofQ(5, q) Using One SubquadrangleQ(4,q )

Leen Brouns, Joseph A. Thas, Hendrik van Maldeghem
<span title="">2002</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/54t3hgai4fhhthc74mj7z7tapu" style="color: black;">European journal of combinatorics (Print)</a> </i> &nbsp;
Let be a finite generalized quadrangle of order (q, q 2 ), and suppose that it has a subquadrangle isomorphic to Q(4, q). We show that is isomorphic to the classical generalized quadrangle Q (5, q) if at least one of the following holds: (1) all linear collineations of extend to ; (2) all subtended ovoids are classical (and we present a uniform proof independent of the characteristic). Further, for q odd, we prove that if every triad {x, y, z} of is 3-regular in and {x, y, z} ⊥⊥ ⊂ , then is
more &raquo; ... sical. We also show that, if for every centric triad {x, y, z} of an ovoid O of the quadrangle ∼ = Q(4, q), q odd, all points of {x, y, z} ⊥⊥ belong to O, then O is classical. L. Brouns et al. An ovoid of the projective space PG(3, q), q > 2, is a set of q 2 + 1 points of PG(3, q), no three of which are collinear. An ovoid of PG(3, 2) is a set of five points no four of which are coplanar. Let be a subquadrangle of the generalized quadrangle . A group G acting on extends to , if for all automorphisms α ∈ G, there is at least one automorphism β acting on such that the restriction of β to is exactly α. A thick finite classical generalized quadrangle is, by definition, one of the following: • the quadrangle arising from a non-singular Hermitian variety in PG(4, q 2 ), denoted by H (4, q 2 ) and of order (q 2 , q 3 ); • the quadrangle arising from a non-singular Hermitian variety in PG(3, q 2 ), denoted by H (3, q 2 ) and of order (q 2 , q); • the quadrangle arising from a non-singular elliptic quadric in PG(5, q), denoted by Q(5, q) and of order (q, q 2 ); it is the dual of H (3, q 2 );
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/eujc.2001.0554">doi:10.1006/eujc.2001.0554</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vaiy4j5bobfh3a32ydyuvywp4a">fatcat:vaiy4j5bobfh3a32ydyuvywp4a</a> </span>
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