We consider geometries belonging to the following diagrams: , , where q is an integer greater than 1 and q denotes orders whereas the integers 0,1,2... denotes types. Let Γ be a geometry in (Cn)q. We prove that, if n > 3 and Aut(Γ) is transitive on the set of elements of type n - 4, the Γ is a building. We show that,, when n⩾3, the geometry Γ is a building iff the property (LL) of [10, Section 6] holds in all residues of Γ of type Finally, we prove that Γ is 2-connected if n⩽4. Let Γ be a geometry in (F4)q. We show that Γ is a building if Aut(Γ) is transitive on the set of points and on the set of hyperlines of Γ. The geometry Γs is a buildingiff the property (LH) of [10, Section 6] holds in it. And we prove that Γ is 2-connected if the property (LL) holds in it.
Pasini, A. (1986). On certain geometries of type Cn and F4. DISCRETE MATHEMATICS, 58(1), 45-61 [10.1016/0012-365X(86)90185-8].
On certain geometries of type Cn and F4
PASINI, ANTONIO
1986-01-01
Abstract
We consider geometries belonging to the following diagrams: , , where q is an integer greater than 1 and q denotes orders whereas the integers 0,1,2... denotes types. Let Γ be a geometry in (Cn)q. We prove that, if n > 3 and Aut(Γ) is transitive on the set of elements of type n - 4, the Γ is a building. We show that,, when n⩾3, the geometry Γ is a building iff the property (LL) of [10, Section 6] holds in all residues of Γ of type Finally, we prove that Γ is 2-connected if n⩽4. Let Γ be a geometry in (F4)q. We show that Γ is a building if Aut(Γ) is transitive on the set of points and on the set of hyperlines of Γ. The geometry Γs is a buildingiff the property (LH) of [10, Section 6] holds in it. And we prove that Γ is 2-connected if the property (LL) holds in it.
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https://hdl.handle.net/11365/7103
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