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Extra info for Combinatorics ’81 in honour of Beniamino Segre, Proceedings of the International Conference on Combinatorial Geometrics and their Applications

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Some of them can be proved for perpendicularity groups and some others have a core which i s valid in all perpendicularity groups. I give two examples. Let a,b,c,d be a chain of perpendiculars with vertices ri,B,C and Ola,d; Hjelmslev says: If b ' is any line joining A,B, then the fourth reflection line d' of the concurrent lines d , a , ( O , b ' ) is incident with J. (Fig. ) If a point C and a line we can draw the perpendicular erect i n L" u (C,u) the perpendicular on This is the line the C-FaiaZZeZ are given, and C d' (C,u).

Buekenhout sul l e ovali pascaliane, BoZZ. UMI, ( 5 ) 18-B (1981). W. Nolte, Gruppen m i t Involutionen, welche Quadriken bestimmen, Arch. , (Basel) 33 (1980), 177-183. F. Rigby, Pascal Ovals i n P r o j e c t i v e Planes, Cmad. J . , 21 (1969), 1462-1 476, B. Segre and G. Korchm&ros, Una p r o p r i e t i i d e g l i insiemi d i p u n t i d i un piano d i Galois c a r a t t e r i z z a n t e q u e l l i formati dai p u n t i d e l l e s i n g o l e r e t t e esterne ad una conica, Rend. Naz. Lincei, (8) 52 (1977), 363-369.

Consequently t h e r e e x i s t ( u . )> 1J 1J such t h a t i s a non-square. 31 Now ifx ! = c . x ? a r e t h e equations d e f i n i n g t h e c o l l i n e a t i o n v o f N, by 1 1 1 2 s e t t i n g ( u . ) ' ~= ( u ! ) , ( v . ) " = ( v i j ) , u31 = x and v31 = -(I/3)y2, we f i n d 1J 1J 1J 0 u' = ( -ll 2 and v i l =-(1/3)(y c2c1) 31 x c c 2 . Therefore o(z(H) Z(H)) >q by lemma On some translation planes admittinga Frobenius group 7. 2) and lemma 4: hence By applying lemma 1, we see a t l a s t that, i f W E S, W # V , H = c(H)xK.

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