The groups of projectivities ofB-ovals

✍ Scribed by Burkard Polster

• Publisher: Springer
• Year: 1992
• Tongue: English
• Weight: 1002 KB
• Volume: 41
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00147457

✦ Synopsis

It is known that the group of projectivities G of a B-oval B = (M, F) is triplytransitive on the set M. Using the classification of all finite triply-transitive groups we list the possible groups of projectivities of the finite B-ovals. Furthermore, we give a definition of hyperbolic parts that covers hyperbolic parts derived from finite B-ovals of odd order. We also list the possible groups of projectivities of finite hyperbolic parts. Following this we define algebraic B-ovals and Lie B-ovals and show that both classes contain only B-conics. Finally, we investigate real B-ovals.

1. Introduction

1.1. A projective oval 0 in an abstract projective plane P is a set of points such that no three are collinear and through each point there passes one and only one line that contains no other point in O. The lines in P that intersect O in no, one or two points are called exterior lines, tangent lines and secant lines respectively.

The classical examples for projective ovals are the projective conics (i.e. irredncible conics in Pappian planes).

1.2. The Buekenhout ovals (B-ovals) are a generalization of this concept. A B-oval B = (M, F) consists of a set M of IMI > 2 points and a set F of permutations of M (called involutions) such that:

(a) each f ~ F has order at most 2; (b) F is quasi sharply 2-transitive in the sense that for any (al, a2), (bl, b2)~M×M, with ai~fib j (i, j=l, 2), there is a unique f~F such that f(ax) = a2, f(bl) = b2. This is the definition given in [8], [16]. 1.3. The natural way to derive B-ovals from projective ovals is described in [5, p. 336, Prop. 1.2]: If O is a projective oval in a projective plane P, we associate to every point p ~ P -O an involutive permutation i t of O such that given any point b in O, b is fixed by ip if the line through b and p intersects 0 only in the point b. The involution i t exchanges b and c ~ 0 if the line through b and p intersects O in the two distinct points b and c. We then define a B-oval B = (M, F), where M --O and F is the set of all involutions i t.

Every B-oval that arises in this manner from a projective oval is called a projective B-oval.