52 research outputs found
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
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
Central aspects of skew translation quadrangles, 1
Modulo a combination of duality, translation duality or Payne integration, every known finite generalized quadrangle except for the Hermitian quadrangles H(4, q2), is an elation generalized quadrangle for which the elation point is a center of symmetry-that is, is a "skew translation generalized quadrangle" (STGQ). In this series of papers, we classify and characterize STGQs. In the first installment of the series, (1) we obtain the rather surprising result that any skew translation quadrangle of finite odd order (s, s) is a symplectic quadrangle; (2) we determine all finite skew translation quadrangles with distinct elation groups (a problem posed by Payne in a less general setting); (3) 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); and (4) we show that finite "generic STGQs," a class of STGQs which generalizes the class of the previous item (but does not contain it by definition), have the expected parameters. We conjecture that the classes of (3) and (4) contain all STGQs
On collineations and dualities of finite generalized polygons
In this paper we generalize a result of Benson to all finite generalized polygons. In particular, given a collineation theta of a finite generalized polygon S, we obtain a relation between the parameters of S and, for various natural numbers i, the number of points x which are mapped to a point at distance i from x by theta. As a special case we consider generalized 2n-gons of order (1,t) and determine, in the generic case, the exact number of absolute points of a given duality of the underlying generalized n-gon of order t
On the mathematical foundations of mutually unbiased bases
In order to describe a setting to handle Zauner's conjecture on mutually unbiased bases (MUBs) (stating that in C-d, a set of MUBs of the theoretical maximal size d + 1 exists only if d is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs. Partial answers will be provided in particular cases; more specifically, we will analyze MUBs with associated operator groups that have nilpotence class 2, and consider MUBs of height 1. We will also confirmZauner's conjecture forMUBswith associated finite nilpotent operator groups
Bol quasifields
In the context of configurational characterisations of symmetric projective
planes, a new proof of a theorem of Kallaher and Ostrom characterising planes of even order of Lenz-Barlotti type IV.a.2 via Bol conditions is given. In contrast to their proof,we need neither the Feit-Thompson theorem on solvability of groups of odd order, nor Bender’s strongly embedded subgroup theorem, depending rather on Glauberman’s Z*-theorem
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