3,928 research outputs found
Classification of skew translation generalized quadrangles, I
We describe new classification results in the theory of generalized quadrangles (= Tits-buildings of rank 2 and type B-2), more precisely in the (large) subtheory of skew translation generalized quadrangles ("STGQs"). Some of these involve, and solve, long-standing open problems
Central aspects of skew translation quadrangles, I
Except for the Hermitian buildings , up to a combination
of duality, translation duality or Payne integration, every known finite
building of type satisfies a set of general synthetic
properties, usually put together in the term "skew translation generalized
quadrangle" (STGQ). In this series of papers, we classify finite skew
translation generalized quadrangles. In the first installment of the series, as
corollaries of the machinery we develop in the present paper, (a) we obtain the
surprising result that any skew translation quadrangle of odd order is
a symplectic quadrangle; (b) we determine all skew translation quadrangles with
distinct elation groups (a problem posed by Payne in a less general setting);
(c) we develop a structure theory for root-elations of skew translation
quadrangles which will also be used in further parts, and which essentially
tells us that a very general class of skew translation quadrangles admits the
theoretical maximal number of root-elations for each member, and hence all
members are "central" (the main property needed to control STGQs, as which will
be shown throughout); (d) we solve the Main Parameter Conjecture for a class of
STGQs containing the class of the previous item, and which conjecturally
coincides with the class of all STGQs.Comment: 66 pages; submitted (December 2013
Singer quadrangles
[no abstract available
A question of Frohardt on -groups, and skew translation quadrangles of even order
We solve a fundamental question posed in Frohardt's 1988 paper [Fro] on
finite -groups with Kantor familes, by showing that finite groups with a
Kantor family having distinct members such that is a central subgroup of and the
quotient is abelian cannot exist if the center of has
exponent and the members of are elementary abelian. In a
similar way, we solve another old problem dating back to the 1970s by showing
that finite skew translation quadrangles of even order are always
translation generalized quadrangles.Comment: 10 pages; submitted (February 2018
Domesticity in generalized quadrangles
An automorphism of a generalized quadrangle is called domestic if it maps no chamber, which is here an incident point-line pair, to an opposite chamber. We call it point-domestic if it maps no point to an opposite one and line-domestic if it maps no line to an opposite one. It is clear that a duality in a generalized quadrangle is always point-domestic and linedomestic. In this paper, we classify all domestic automorphisms of generalized quadrangles. Besides three exceptional cases occurring in the small quadrangles with orders (2, 2), (2, 4), and (3, 5), all domestic collineations are either point-domestic or line-domestic. Up to duality, they fall into one of three classes: Either they are central collineations, or they fix an ovoid, or they fix a large full subquadrangle. Remarkably, the three exceptional domestic collineatons in the small quadrangles mentioned above all have order 4
Elation generalised quadrangles of order (s,p), where p is prime
We show that an elation generalised quadrangle which has p+1 lines on each
point, for some prime p, is classical or arises from a flock of a quadratic
cone (i.e., is a flock quadrangle).Comment: 14 page
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