Fahnentransitive Automorphismengruppen von Blockplänen. (Flag transitive automorphism groups of block designs.) (Q1065354)

The following theorem is the main result of the paper. Theorem 1. Let \({\mathcal B}\) be a block design on a set X with \(\lambda =1\) and G a flag transitive group of automorphisms of \({\mathcal B}\) such that each two point stabilizer \(G_{rs}\) acts trivially on the unique block determined by r and s. Let \(x\in X\) and let A be a nontrivial normal p-subgroup of \(G_ x.\) Then each one point stabilizer \(G_ r\) contains exactly one conjugate \(A^ r\) of A and the family of fixed point sets \(\{Fix(A^ r\cap G_ s)| \quad r,s\in X,\quad r\neq s\}\) forms a block design on X on which G acts as a group of automorphisms. Theorem 1 is needed to prove the following result. Theorem 2. Let X and G as in Theorem 1. Let \(x\in X\) and suppose that \(G_ x\) has an abelian normal subgroup which is not semiregular on X- \(\{\) \(x\}\). Then for some integer \(n\geq 3\) and some prime power q, we have \(PSL(n,q)\leq G\leq P\Gamma L(n,q).\) Theorem 2 generalizes Theorem A of the paper of \textit{M. O'Nan} in Math. Z. 127, 301-314 (1972; Zbl 0258.20003).