2-transitive abstract ovals of odd order

Aequatlones Mathematlcae 33 (1987) 208-219 Umverstty of Waterloo

0001 9054/87/002208-125150 + 0 20/0 © 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 ( a O = a2, f ( b t ) = b2. In this paper we shall be concerned 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 e a c h f E ~ . 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

Vol 33, 1987

2-transitive abstract ovals of odd order

209

An abstract oval ts called 2-transmve ff tt has an automorphism group actmg 2transittvely on M. An involution f e ~ is called regular tf tt is an automorphism. Clearly, all involutions of the classical abstract oval are regular The study of regular involutions and automorphism groups generated by them is a powerful tool in the study of abstract ovals Buekenhout's paper [3] has been the basis for regular revolutions His stimulating and penetrating mvesttgations were followed by numerous authors. Details cannot be given here, the reader should refer to [7], [16] We mention only Theorem 8 8 of [3] which states that, if all involutions of an abstract oval of even order are regular, then it ts the classtcal one Thts result was extended to any odd order in [15] In a recent paper [8] we were concerned with regular involutions of 2-transitive abstract ovals of even order We have an almost complete classification in which only a few gaps remain It is in the case of even order that the picture is very rich and group theory is especmlly powerful For the case of odd order, the classtcal ones are the only known 2-transitive abstract ovals If no other example extsts, then all Involutions of any 2-transitive abstract oval of odd order are regular The author beheves that the classtcal one is the untque 2-transitive abstract oval of odd order Our Theorem 2 states this for n = 3 (mod 4) The case n = 1 (mod 4) is more complicated in our context because lnvolutorial automorphisms with x/~ + 1 fixed pomts can be involved In an effort to investigate this case, we give a proof under the assumption that the abstract oval admits some regular involutions, see Theorem 3. Moreover, a proof without any additional hypothesis is given by using the classification of the finite simple groups

2. Notation and preliminary results Fairly standard notation ts used A certam famtharlty with abstract ovals as well as with finite groups is assumed. The reader should refer to [3], [7], [13], [16], [18]. Throughout this paper, B = (M, ~ ) denotes an abstract oval of odd order n We shall assume the following elementary results on finite abstract ovals; for the proofs see [3]. consists ofn z involutions They are permutations of order two Each involution has either 2 or 0 fixed points, and it ts called a hyperbolic or elliptic mvolution, respectively. Let E denote the set of all elhptlc involutions Then Igl = n(n - 1)/2 For any two &stlnct points a, b of M, let E(a, b) denote the set of all eUlptlc Involutions which exchange a with b. Then IE(a, b)l = (n - 1)/2. For hyperbolic revolutions, let H and H(a,b) be defined similarly Then IHI = n(n + 1)/2,

210

G KORCHMAROS

AEQ MATH

IH(a, b)l = (n - 1)/2. Moreover, for any point a of M, let H(a) denote the set of all revolutions fixing a. Then In(a)l = n. For any two distinct points a, b of M, I n ( a ) n n(b)l = 1. For any automorphlsm 0, let # denote the m a p f---) gfo-L Then # maps ~- onto ~tself. So, any automorphism group G mduces a permutation group (7 on ~ . G and (7 are ~somorphic. (7 ~s not transmve, as it maps E and H onto ~tself. Assume that g has order two, and denote by M ( g ) the set of all fixed points of # Then IM(g)l = 0, 2 or x/n + 1. If IM(g)l -- 2, then # is a regular revolution. This result is not true in general for IM(g)l = 0 If IM(o)I = ~ + 1, then ~(f) = f f e ~-, lmphes f ~ H Gwen a finite set M, let B, = (M, ~ , ) (t = 1, 2) be abstract ovals of odd order n such that for a n y f a ~ Ha,f2 ~ E2, the e q u a t l o n f l ( x ) = f2(x), x ~ M, has at most two solutmns Put ~ = H~ u E 2. Then also B = (M, ~-) is an abstract oval Moreover, ff there exists a permutation group G of M which is an automorphlsm group of both B, = (M, ~-,) such that (7 acts transitively on Ha, then the above can be weakened as follows" There exists a f l e Ha such that for any f2 e E2 the equation f l ( x ) = f2(x), x ~ M, has at most two solutions.

3. A characterization of the classical abstract oval Our aim is to prove the following THEOREM 1. L e t B = (M, :~r) denote an abstract oval o f order q = pr wtth odd prime p I f B = (M, ~r) admtts an automorphtsm group G actm9 on M as P S L ( 2 , q) m tts usual 2-transltwe representatton, then B = (M, :~r) ts the classwal abstract oval Under our hypothesis M m a y be identified with GF(q) w { ~ } m such a way that G coincides with PSL(2, q). Let B' = (M, ~ - ' ) denote the classical abstract oval over GF(q). We prove Theorem 1 by proving ~- = ~ ' . Case q = 1 (mod 4) Any a u t o m o r p h l s m g ~ G of order two has two fixed points Therefore g e H. Take any f e E, and let (7y denote the stabihzer o f f In (7. Then IGzl > q + 1 by IGI = IGI = q(q2 _ 1)/2. For any ~ e (7y the corresponding g ~ G is either without fixed points, or it has an even number of fixed points. Since each element of order p of G has a unique fixed pomt, p X 1(7sl follows. Let K = {1, gl, 02, 93} be any Klein subgroup of G Each g, has two fixed points, let a,~ (j = 1, 2) denote them. Then a,j ~ a,v when (t,j) ~ (u, v), 1 ~< t, u ~< 3, and 1 ~<j, v