By Buekenhout F., Huybrechts C.

We end up the lifestyles of a rank 3 geometry admitting the Hall-Janko staff J2 as flag-transitive automorphism workforce and Aut(J2) as complete automorphism staff. This geometry belongs to the diagram (c·L*) and its nontrivial residues are entire graphs of dimension 10 and twin Hermitian unitals of order three.

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32. Let E = {1, 2, . . , n} and C consist of all four-point subsets of E. Prove that C is the set of circuits of a planar geometry on E. Is it a figure? If it were we could call it a figure of n coplanar points in general position. 33. Let E = {0, 1, 2, . . , 6} and let K = {{0, 1, 3}, {1, 2, 4}, {2, 3, 5}, {3, 4, 6}, {4, 5, 0}, {5, 6, 1}, {6, 0, 2}}. Suppose that C = K ∪ {C ⊂ E : |C| = 4, C contains no member of K}. Prove that C is the set of circuits of a planar geometry on E. 34. 33 the Fano plane, or planar Fano geometry, after the 20th Century Italian geometer Gino Fano.

Thus a ∨ b ∨ c = x ∨ y ∨ z. Moreover, if w ∈ x ∨ y then either w ∈ {x, y} or {x, y, w} is a circuit. This guarantees that w ∈ x ∨ y ∨ z. Hence x ∨ y ⊂ x ∨ y ∨ z = a ∨ b ∨ c. 22. In any combinatorial geometry, let p be any point, and L any line not containing p. Then there is exactly one plane of the geometry containing both p and L. We denote this plane by p ∨ L or L ∨ p. P. If L = a ∨ b, then P = a ∨ b ∨ p. The next few results show that the intersections of pairs of planes and lines are satisfyingly simple.

7, we prove that the choice of any three non-collinear points of a plane is adequate for its definition. The points a, b,and c are called “naming points” for a plane a ∨ b ∨ c in a geometry . 17. If x, y, and z are any three non-collinear points of a plane P of a combinatorial geometry, then P = x ∨ y ∨ z. In other words, a plane can be “named” by any three of its points that are non-collinear. P. We claim that given three naming points for a plane, any point of the plane that is not on a line determined by two of the naming points may be used to replace the third naming point when describing the plane.