Elation and translation semipartial geometries

Abstract

We introduce a theory of elation and translation semipartial geometries (SPG). Starting from an SPG-family (G, J ), i.e. a not necessarily abelian group G and a collection of subgroups J = {S 0 , . . . , S t } satisfying some extra condition, we construct a semipartial geometry S as a coset geometry. We show that there are strong relations between the theory of these geometries and that of elation and translation generalized quadrangles. We show for example that the theory of translation semipartial geometries is in fact almost equivalent to the study of SPG-reguli in PG(n, q). We introduce a special class of automorphisms, called parallelisms, for these geometries and examine the structure of fixed points and lines under these automorphisms. In the case that G is abelian we show that in almost all cases Aut(S) A L(n + 2, q) for certain n and q.

Partial geometries pg (s, t, 2) with an abelian Singer group and a characterization of the van Lint-Schrijver partial geometry

“…. , S t }) is an SPG-family in the sense of [6]. From Theorem 2.5 of that same paper [6] it then follows that S is isomorphic to a partial geometry constructed from a PG-regulus.…”

Generalized Quadrangles with an Abelian Singer Group

“…Under a small restriction this is indeed possible. The basic ideas for the proof of this observation are taken from De Winter [4] (see also De Clerck et al [2]). …”

Distance-Regular (0,α)-Reguli

“…Distance-regular geometries arising from distance-regular (0, α)-reguli can be seen as translation (0, α)-geometries, which are a slight generalisation of the translation semipartial geometries described in [8]. The latter in their turn extend the concept of translation generalised quadrangles, see [12].…”

“…In this case R is an SPG-regulus and the corresponding geometry S(R) is a semipartial geometry. In [8], De Winter proves that if t 4 = α ∈ {1, q m+1 − 1, q m+1 }, then R is α-geometric, α = q (m+1)/2 , and S(R) is isomorphic to the semipartial geometry T * N (B) arising from a Baer subspace in PG(N, q m+1 ), with N = (n + 1)/(m + 1) − 1. This theorem can be generalised as follows.…”