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

Abstract

Let S be a proper partial geometry pg(s, t, 2), and let G be an abelian group of automorphisms of S acting regularly on the points of S. Then either t ≡ 2 (mod s + 1) or S is a pg(5, 5, 2) isomorphic to the partial geometry of van Lint and Schrijver (Combinatorica 1 (1981), 63-73). This result is a new step towards the classification of partial geometries with an abelian Singer group and further provides an interesting characterization of the geometry of van Lint and Schrijver.

Proper partial geometries with Singer groups and pseudogeometric partial difference sets

“…If there are no subgroup lines, the geometry is said to be of rigid type. The following direct consequence of Lemma 16(c) was proved in [10].…”

“…If a proper pg(s + 1, t + 1, 2) with an abelian Singer group G of spread type exists, then |G| = (s + 1) 3 , t = 2(s + 2), and G is an elementary abelian 3-group by [10,Theorem 4.1]. Furthermore, by [10, Corollary 6.12], the only pg(s + 1, t + 1, 2) with an abelian Singer group of rigid type is the Van Lint-Schrijver geometry [11].…”

“…In the first case, we call L i a subgroup line, in the second case an orbit line. Following [10], we say the partial geometry is of spread type if all L i 's are subgroup lines. If there are no subgroup lines, the geometry is said to be of rigid type.…”

“…De Winter [10,Remark 5.3] conjectured that no pg(s + 1, t + 1, 2) with abelian Singer group of mixed type exists. We will verify this conjecture except for one exceptional parameter family by proving the following.…”

Automorphisms of strongly regular graphs with applications to partial difference sets

“…This result has turned out to be useful at various occasions in the theory of generalized quadrangles, see for example [3,12]. In 2006 the first author [2] generalized this theorem to partial geometries and used it to obtain a characterization of the socalled Van Lint-Schrijver partial geometry. In 2010 Temmermans in her dissertation (see also [13]) provided further generalizations for other geometries, including partial quadrangles, and used these generalizations to study polarities of the geometries under investigation.…”