Triple factorisations of the general linear group and their associated geometries

Abstract

Triple factorisations of finite groups G of the form G = P QP are essential in the study of Lie theory as well as in geometry. Geometrically, each triple factorisation G = P QP corresponds to a G-flag transitive point/line geometry such that 'each pair of points is incident with at least one line'. We call such a geometry collinearly complete, and duality (interchanging the roles of points and lines) gives rise to the notion of concurrently complete geometries. In this paper, we study triple factorisations of the general linear group GL(V ) as P QP where the subgroups P and Q either fix a subspace or fix a decomposition of V as V 1 ⊕ V 2 with dim(V 1 ) = dim(V 2 ).

Small maximal partialt-spreads inPG(2t+1,q)Boeck1 2015European Journal of Combinatorics0000View full textAdd to dashboardCite

“…The bound presented in Theorem 3.6 improves the bound q + √ q − 1 from [3] for all q. It improves the bound 5 from [1] for all q ≥ 3. For q = 2, the lower bound 5 is still the best lower bound.…”

“…It improves on the lower bound q + √ q − 1, which was proved in [3,Theorem 4] (actually, the bound proved by Beutelspacher in [3] was a little bit better). Very recently, the lower bound 5 (which is useful when q = 2) was proved in [1,Lemma 4.15]. For t = 1, this problem was already studied by Glynn.…”

A generalisation of Johnson graphs with an application to triple factorisations

“…This motivated Alavi and Burness [3] to study large maximal subgroups H of finite simple groups G (i.e., |H| ≤ |G| 1/3 ). In this direction, various triple factorisations of general linear groups GL(V ) have been studied (see [1,2]). Triple factorisations Sym(Ω) = ABA of symmetric groups with A and B conjugate subgroups have been studied in [8] and Theorem 1.1 focuses on intransitive factorisations of Sym(Ω).…”

“…The geometries associated to (2,2,2) are simply the generalised di-gons, and their automorphism groups G give degenerate factorisations G = AB. The geometries associated to (3,3,3) and (3,3,4) are flag-transitive linear spaces which have been classified up to the one-dimensional affine case [6].…”

“…If Y is an edge between X and Z , then Y is a k-subset of Ω satisfying (2). Set T := X ∩ Y ∩ Z with t := |T |, and define Fig.…”