Aequatlones Mathematlcae 33 (1987) 208-219 0001 9054/87/002208-125150 + 0 20/0 Umverstty of Waterloo © 1987 Blrkhauser Verlag, Basel 2-transitive abstract ovals of odd order G. KORCHMAROS I. Introduction Following Buekenhout [3,1, [4,1, an abstract oval (called also free oval, or Buekenhout oval, or for brevity B-oval) B = (M, ~) is defined as a set M of elements, called points, together with a sharply quasi 2-transmve set ~ of mvolutonal permutations of M, called mvoluttons. Here sharply quast 2-transitivity means that for any two pairs of points (a t, a2), (b~, b 2) with a, :# b~ 0,J = 1, 2) there exists a unique element f~ ~- such that f(aO = a2, f(bt)= b2. In this paper we shall be con- cerned with fintte abstract ovals We define the order n of a finite abstract oval by n = IMI- 1 The classwal abstract oval arises from the linear group PGL(2, q), q = p" and p prime, regarded m its 3-transmve representation over GF(q)" M = GF(q) w {~}, and ~ consists of all elements of order 2 of PGL(2, q) with the addmon, for p = 2, ~ts identity element It ~s interesting that there is a natural way of denwng abstract ovals from projective ovals, the classical one arises from the irreducible comc of PG(2, q). There are known, however, abstract ovals not obtainable m such a way. We do not discuss these here, the reader is referred to [5], [6-1, [7-1, [20,1. An automorphlsm g of an abstract oval is a permutation of M which reduces a map of ~ onto itself, I e. g(f) = gfg-t ~ ~ for eachfE ~. The full automorphlsm group of the classmal abstract oval is PFL(2, q), and this property characterizes it. A weaker result is given m this paper. Our Theorem 1 shows that the classmal abstract oval is characterized as the only abstract oval w~th an automorphlsm group acting on M as PSL(2, q) m its usual 2-transitwe representahon Notice that for p = 2 this characterization was given by Buekenhout [3]. AMS (1980) subject classification Primary 51T20, 51T99 Secondary 12K05,20B10 Manuscrtpt recewedNovember 14 1985