A question of Frohardt on 22-groups, and skew translation quadrangles of even order

We solve a fundamental question posed in Frohardt's 1988 paper [Fro] on finite $2$-groups with Kantor familes, by showing that finite groups with a Kantor family $(\mathcal{F},\mathcal{F}^*)$ having distinct members $A, B \in \mathcal{F}$ such that $A^* \cap B^*$ is a central subgroup of $H$ and the quotient $H/(A^* \cap B^*)$ is abelian cannot exist if the center of $H$ has exponent $4$ and the members of $\mathcal{F}$ 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 $(t,t)$ are always translation generalized quadrangles.Comment: 10 pages; submitted (February 2018

Central aspects of skew translation quadrangles, I

Except for the Hermitian buildings $\mathcal{H}(4,q^2)$, up to a combination of duality, translation duality or Payne integration, every known finite building of type $\mathbb{B}_2$ 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 $(s,s)$ 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

Unextendible mutually unbiased bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)

We consider questions posed in a recent paper of Mandayam et al. (2014) on the nature of unextendible mutually unbiased bases. We describe a conceptual framework to study these questions, using a connection proved by the author in Thas (2009) between the set of nonidentity generalized Pauli operators on the Hilbert space of N d-level quantum systems, d a prime, and the geometry of non-degenerate alternating bilinear forms of rank N over finite fields F d We then supply alternative and short proofs of results obtained in Mandayam et al. (2014), as well as new general bounds for the problems considered in loc. cit. In this setting, we also solve Conjecture 1 of Mandayam et al. (2014) and speculate on variations of this conjecture

Hyperfield extensions, characteristic one and the Connes-Consani plane connection

Inspired by a recent paper of Alain Connes and Catherina Consani which connects the geometric theory surrounding the elusive field with one element to sharply transitive group actions on finite and infinite projective spaces ("Singer actions"), we consider several fudamental problems and conjectures about Singer actions. Among other results, we show that virtually all infinite abelian groups and all (possibly infinitely generated) free groups act as Singer groups on certain projective planes, as a corollary of a general criterion. We investigate for which fields $\mathbb{F}$ the plane $\mathbf{P}^2(\mathbb{F}) = \mathbf{PG}(2,\mathbb{F})$ (and more generally the space $\mathbf{P}^n(\mathbb{F}) = \mathbf{PG}(n,\mathbb{F})$) admits a Singer group, and show, e.g., that for any prime $p$ and any positive integer $n > 1$, $\mathbf{PG}(n,\overline{\mathbb{F}_p})$ cannot admit Singer groups. One of the main results in characteristic $0$, also as a corollary of a criterion which applies to many other fields, is that $\mathbf{PG}(m,\mathbb{R})$ with $m \ne 0$ a positive even integer, cannot admit Singer groups.Comment: 25 pages; submitted (June 2014). arXiv admin note: text overlap with arXiv:1406.544