On generalized quadrangles with a point regular group of automorphisms

Abstract

A generalized quadrangle is a point-line incidence geometry such that any two points lie on at most one line and, given a line ℓ and a point P not incident with ℓ, there is a unique point of ℓ collinear with P . We study the structure of groups acting regularly on the point set of a generalized quadrangle. In particular, we provide a characterization of the generalized quadrangles with a group of automorphisms acting regularly on both the point set and the line set and show that such a thick generalized quadrangle does not admit a polarity. Moreover, we prove that a group G acting regularly on the point set of a generalized quadrangle of order (u 2 , u 3 ) or (s, s), where s is odd and s + 1 is coprime to 3, cannot have any nonabelian minimal normal subgroups.2010 Mathematics Subject Classification. Primary 51E12, 05B25.

Distance-regular Cayley graphs with small valency

“…We note that this result also follows from more extensive results by Bamberg and Giudici [5, Theorem 1.1] and by Swartz [26,Theorem 1.3]. We remark that also the result that Tutte's 8-cage -the incidence graph of the unique generalized quadrangle of order (2, 2) -is not a Cayley graph, can be obtained using the point graph.…”

“…It follows that there is an index 2 subgroup G that acts regularly on both the point set and on the line set, as an automorphism group of the generalized polygon. This situation has been studied by Swartz [26] for generalized quadrangles. Using results by Yoshiara [29] (who exploited an idea of Benson [9]; cf.…”

“…Using results by Yoshiara [29] (who exploited an idea of Benson [9]; cf. [23, 1.9.1]), Swartz [26] showed that s + 1 must be coprime to 2 and 3. Consequently, we have the following result.…”

The point regular automorphism groups of the Payne derived quadrangle of $W(q)$

“…Ghinelli [13] was the first to study the generalized quadrangles admitting a point regular group, where she used representation theory and difference sets to study the case where the generalized quadrangle has order (s, s) with s even. Further progress was made in [31] and [27]. For instance, it is shown in [31] that finite thick generalized quadrangles of order (t 2 , t) does not admit a point regular automorphism group.…”

“…By combining the results in [14] and [31], it was shown in [4] that any skew-translation generalized quadrangle of order (q, q), q odd, is isomorphic to the classical symplectic quadrangle W (q). Swartz [27] initiated the study of generalized quadrangles admitting an automorphism group that acts regularly on both points and lines. Up till 2011, all the known finite generalized quadrangles admitting a point regular group arise by Payne derivation from a thick elation quadrangle Q of order (s, s) with a regular point, and their point regular groups are induced from the elation groups of Q.…”