The classification of the finite simple groups has shown that most of the finite simple groups possess a natural representation as a flag transitive automorphism group of a suitable geometry; conversely, a successful treatment of larger classes of geometries can often be ensured only under additional algebraic hypotheses, for instance the existence of an automorphism group with certain transitivity properties. We thus may ask whether a flag transitive automorphism group of a finite geometry is again (more or less) simple or, on the contrary, may have an enriched normal structure.
If the underlying geometry is strongly determined by the structure of a simple group, one should expect the question to be answered in the affirmative: G. M. Seitz [10; Theorem A] has shown that, apart from a few exceptions, F*(G)$F*(C) whenever G is a flag transitive automorphism group of a building 2 which belongs to a finite Chevalley group C of normal or twisted type.
Assume that 2 is of type A n with 2 n. Then we may identify the automorphisms of 2 naturally with those of the 2-design D defined by 2 and consisting of the points and hyperplanes of 2. As the flags of D are precisely the flags of type [1, n] of 2, a flag transitive automorphism group of 2 certainly acts flag transitively also on D. It is remarkable that already this weaker flag transitivity yields the conclusion of Seitz's result (in the case A n , 2 n). More precisely, for each integer n with 2 n, for each prime power q, and for each flag transitive automorphism group G of a 2-design isomorphic to P n&1 (n, q), we have F*(G)$PSL n+1 (q), provided that (n, q) ร [(2, 2), (2, 8), (3, 2)].
With regard to the question brought up in the beginning this result fits in a more general framework. For the parameters r and * of a 2-design isomorphic to P n&1 (n, q), we have r=q n&1 + } } } +q+1 and *= q n&2 + } } } +q+1; in particular, r and * are coprime. On the other hand, article no. 0049 102