Isomorphisms of groups related to flocks

A truly fruitful way to construct finite generalized quadrangles is through the detection of Kantor families in the general 5-dimensional Heisenberg group H2(q) over some finite field Fq . All these examples are so-called “flock quadrangles”. Payne (Geom. Dedic. 32:93–118, 1989) constructed from the Ganley flock quadrangles the new Roman quadrangles, which appeared not to arise from flocks, but still via a Kantor family construction (in some group G of the same order as H2(q)). The fundamental question then arose as to whether H2(q) ∼= G (Payne in Geom. Dedic. 32:93–118, 1989). Eventually the question was solved in Havas et al. (Finite geometries, groups, and computation, pp. 95–102, de Gruyter, Berlin, 2006; Adv. Geom. 26:389–396, 2006). Payne’s Roman construction appears to be a special case of a far more general one: each flock quadrangle for which the dual is a translation generalized quadrangle gives rise to another generalized quadrangle which is in general not isomorphic, and which also arises from a Kantor family. Denote the class of such flock quadrangles by C . In this paper, we resolve the question of Payne for the complete class C . In fact we do more—we show that flock quadrangles are characterized by their groups. Several (sometimes surprising) by-products are described in both odd and even characteristic.