In this note we characterize thick finite generalized quadrangles constructed from a generalized hyperoval as those admitting an abelian Singer group, i.e., an abelian group acting regularly on the points.

Let S be a proper partial geometry pg(s, t, 2), and let G be an abelian group of automorphisms of S acting regularly on the points of S. Then either t ≡ 2 (mod s + 1) or S is a pg(5, 5, 2) isomorphic to the partial geometry of van Lint and Schrijver (Combinatorica 1 (1981), 63-73). This result is a new step towards the classification of partial geometries with an abelian Singer group and further provides an interesting characterization of the geometry of van Lint and Schrijver.

Let $\Pi = (P,L,I)$ denote a rank two geometry. In this paper, we are interested in the largest value of $|X||Y|$ where $X \subset P$ and $Y \subset L$ are sets such that $(X \times Y) \cap I = \emptyset$. Let $\alpha(\Pi)$ denote this value. We concentrate on the case where $P$ is the point set of $\mathsf{PG}(n,q)$ and $L$ is the set of $k$-spaces in $\mathsf{PG}(n,q)$. In the case that $\Pi$ is the projective plane $\mathsf{PG}(2,q)$, where $P$ is the set of points and $L$ is the set of lines of the projective plane, Haemers proved that maximal arcs in projective planes together with the set of lines not intersecting the maximal arc determine $\alpha(\mathsf{PG}(2,q))$ when $q$ is an even power of $2$. Therefore, in those cases,\[ \alpha(\Pi) = q(q - \sqrt{q} + 1)^2.\] We give both a short combinatorial proof and a linear algebraic proof of this result, and consider the analogous problem in generalized polygons. More generally, if $P$ is the point set of $\mathsf{PG}(n,q)$ and $L$ is the set of $k$-spaces in $\mathsf{PG}(n,q)$, where $1 \leq k \leq n - 1$, and $\Pi_q = (P,L,I)$, then we show as $q \rightarrow \infty$ that \[ \frac{1}{4}q^{(k + 2)(n - k)} \lesssim \alpha(\Pi) \lesssim q^{(k + 2)(n - k)}.\] The upper bounds are proved by combinatorial and spectral techniques. This leaves the open question as to the smallest possible value of $\alpha(\Pi)$ for each value of $k$. We prove that if for each $N \in \mathbb N$, $\Pi_N$ is a partial linear space with $N$ points and $N$ lines, then $\alpha(\Pi_N) \gtrsim \frac{1}{e}N^{3/2}$ as $N \rightarrow \infty$.

In 1969 Denniston [3] gave a construction of maximal arcs of degree d in Desarguesian projective planes of even order q, for all d dividing q. In 2002 Mathon [8] gave a construction method generalizing the one of Denniston. We will give a new geometric approach to these maximal arcs. This will allow us to count the number of isomorphism classes of Mathon maximal arcs of degree 8 in PG(2, 2 h ), h prime.

In this article we generalize a theorem of Benson (J Algebra 15:443-454, 1970) for generalized quadrangles to strongly regular graphs, deriving numerical restrictions on the number of fixed vertices and the number of vertices mapped to adjacent vertices under an automorphism. We then use this result to develop a few new techniques to study regular partial difference sets (PDS) in Abelian groups. Ma (Des Codes Cryptogr 4:221-261, 1994) provided a list of parameter sets of regular PDS with k ≤ 100 in Abelian groups for which existence was known or had not been excluded. In particular there were 32 parameter sets for which existence was not known. Ma (J Stat Plan Inference 62:47-56, 1997) excluded 13 of these parameter sets. As an application of our results we here exclude the existence of a regular partial difference set for all but two of the undecided parameter sets from Ma's list.

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All known finite generalized quadrangles that admit an automorphism group acting sharply transitively on their point set arise by Payne derivation from thick elation generalized quadrangles of order s with a regular point. In these examples only two groups occur: elementary abelian groups of even order and odd order Heisenberg groups of dimension 3. In [2] the authors determined all generalized quadrangles admitting an abelian group with a sharply transitive point action. Here, we classify thick finite generalized quadrangles admitting an odd order Heisenberg group of dimension 3 acting sharply transitively on the points. In fact our more general result comes close to a complete solution of classifying odd order Singer p-groups.AMS Classifications 05B10 · 05B25 · 05E20 · 20B25 · 51E12 · 51E20 Singer generalized quadranglesA Singer group (w.r.t. points) of a point-line incidence geometry is an automorphism group of the geometry acting sharply transitively on its points. Before proceeding, recall that a finite (thick) 1 generalized n-gon S of order (s, t), n ≥ 2, s > 1, t > 1 (where all the parameters are finite), is a 1 − (v, s + 1, t + 1) design whose incidence graph has girth 2n and diameter n. Whenever s = t the generalized n-gon is said to have order s. The (finite) 1 Only thick generalized n-gons are considered in this note, even when we do not mention this explicitly in statements, etc.