On the Normal Structure of Flag Transitive Automorphism Groups of 2-designs

Let D:= (X,B,I) be a 2-design. For each xeX set Bx:= {jeB:xIj}, and define Xh dually for each h e B. We call D reduced if Xh = Xt implies that h = i for all h,ieB. Assume that D is a reduced 2-design with parameters v, b, k, r, X. Let G be a flag transitive automorphism group of D. By F*(G) we mean (as usual) the generalized Fitting subgroup of G which is defined to be the inverse image in G of the socle of F(G)CG(F(G))/F(G). (For a discussion of the importance of the generalized Fitting subgroup see [1, Chapter 11].) There exist a number of arithmetical conditions concerning the parameters of D which imply that G acts primitively on X; see [4, 2.3.7]. Clearly, in this case, F*(G) is a minimal abelian normal subgroup of G or the direct product of pairwise isomorphic non-abelian simple groups. One of these conditions is (r, X) = 1 in which case one can even show that F*(G) is abelian or simple; see [7]. On the other hand, by [3] or [2, Theorem III] it is possible that G acts imprimitively on X, and it is this case which will be investigated in the present paper. First of all, for each set of imprimitivity X+ of G, we define two flag transitive tactical configurations derived from D, one of them defined on X+, the other one with point set {X%:geG}. These constructions which mostly turn out to be reduced 2designs are of interest for at least two reasons. First, they enable us to prove that \n(F(G))\ < 2. Secondly, whenever we have \n(F(G))\ = 2, they produce a reduced 2design Z)# and a flag transitive automorphism group G+ of D+ with ^(/^G^))! = 2 such that F(G+) acts regularly on the point set of Z>+. We shall show that, if F(G) acts regularly on X and if there exist two different prime numbers p and q and a positive integer e such that \F(G)\ = pq, then (v, k, X) = (45,12,8) and D is unique with respect to these properties. We also construct a reduced 2-design with these parameters and determine all possible choices for G. The following lemma will be needed frequently. For a proof see [4, (2.1.5), 1.3.8].