Generalized quadrangles admitting a sharply transitive Heisenberg group

Abstract

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.

On generalized quadrangles with a point regular group of automorphisms

“…Thus 2 n q 2 2 2n−2 + 2, and so q 2 2 n−2 + 1 2 n−1 , and, since q is a power of 2, q 2 2 n−2 . But then s + 1 = 2 n−1 q 2 2q 4 , contradicting (5). Now suppose that 2 2n−2 − 2 n q 2 + 2 < 0.…”

“…In [4], it is shown that if Q is a thick finite generalized quadrangle of order (s, t) admitting an abelian automorphism group that acts regularly on points, then Q is isomorphic to a generalized quadrangle T * 2 (O) arising from a generalized hyperoval, and the group acting on Q is elementary abelian of order 2 3n for some natural number n. (For further background information and descriptions of the generalized quadrangles referred to in this paragraph, the curious reader is again directed to [13].) In [5], it is shown that if a Heisenberg group of order q 3 , where q is odd, acts regularly on the point set of Q, then Q is a Payne-derived generalized quadrangle of a thick elation generalized quadrangle having a regular point. (A regular point of a generalized quadrangle is defined combinatorially and has no relation to a regular group action.)…”

Point regular groups of automorphisms of generalised quadrangles

“…In [6] the authors prove Theorem 5.1. Let Q be a generalised quadrangle of order (s, t) admitting a point regular group G, where G is a p-group and p is odd.…”

Central aspects of skew translation quadrangles, 1

“…We deduce some structural results from §7.1. The following theorem is essentially contained in [12], and uses the Payne-derived quadrangle P(S x , x). We give a short different proof here using centrality.…”