ON 2-TRANSITIVE GROUPS IN WHICH THE STABILIZER OF TWO POINTS FIXES ADDITIONAL POINTS t

Let F be a 2-transitive group of finite degree v such that the stabilizer T xy of the distinct points x and y fixes precisely k points, where 2 < k < v. The only known non-solvable groups with this property are of the following types: (i) a 2-transitive collineation group of PG(d, 2) for some d; (ii) a 2-transitive collineation group of AG(d, k) for some d; and (iii) PFL(2, 8) in its representation of degree 28. We shall prove some characterizations of groups of these types. Let X be the set of fixed points of T xy and T x = N(T xy) the global stabilizer of X. Our main result (Theorem 5.1) states that F is of the form (i) or (ii) provided that F x n F x is transitive on the points not in X. This generalizes a result of Ito [13] (see [4; pp. 47-48]) which assumes that T xy is transitive and regular on the points not in X. Our approach is, however, quite different from Ito's. When k is even we use a result of Bender [2]. When k is odd a result of Hall [9] is used in an unexpected way. The case where T xy is not regular on the points it moves is deduced from the case where this group is regular by means of a result of Glauberman [7].