Tactical decompositions and orbits of projective groups

AbstractWe consider decompositions of the incidence structure of points and lines of PG(n, q) (n⩾3) with equally many point and line classes. Such a decomposition, if line-tactical, must also be point-tactical. (This holds more generally in any 2-design.) We conjecture that such a tactical decomposition with more than one class has either a singleton point class, or just two point classes, one of which is a hyperplane. Using the previously mentioned result, we reduce the conjecture to the case n=3, and prove it when q2+q+1 is prime and for very small values of q. The truth of the conjecture would imply that an irreducible collineation group of PG(n, q) (n⩾3) with equally many point and line orbits is line-transitive (and hence known)