By A. Barlotti, P.V. Ceccherini and G. Tallini (Eds.)
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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
Example text
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.