Classification of skew translation generalized quadrangles, I

Generalized n-gons were introduced by Tits in a famous work on triality [20] of 1959, in order to propose an axiomatic and combinatorial treatment for semisimple algebraic groups (including Chevalley groups and groups of Lie type) of relative rank 2. They are the central rank 2 incidence geometries, and the atoms of the more general “Tits-buildings.” If the number of elements of a generalized n-gon is finite, a celebrated result of Feit and Higman [1] guarantees that n is restricted to the set {3, 4, 6, 8}. Note that projective planes are nothing else but generalized 3-gons. Generalized 4-gons are also called generalized quadrangles, and, certainly in the finite case, they are considered as being the main players in the class of generalized n-gons, and one of the most studied types of incidence geometries. The most fruitful way to construct finite generalized quadrangles is through a now standard group coset geometry construction, in which a group E provided with certain sets of subgroups E = {Ei | i ∈ I} and E∗ = {E∗ i | i ∈ I}, I an index set, and satisfying some strong intersection properties, is used to represent a generalized quadrangle. Such a system of groups (E , E∗) is called a Kantor family for E, and the defining properties are as follows.