Tactical Decompositions of Steiner Systems and Orbits of Projective Groups

Abstract

Block's lemma states that the numbers m of point-classes and n of block-classes in a tactical decomposition of a 2-(v, k, λ) design with b blocks satisfy m ≤ n ≤ m + b − v. We present a strengthening of the upper bound for the case of Steiner systems (2-designs with λ = 1), together with results concerning the structure of the block-classes in both extreme cases. Applying the results to the Steiner systems of points and lines of projective space PG(N, q), we obtain a complete classification of the groups inducing decompositions satisfying the upper bound; answering the analog of a question raised by Cameron and Liebler (P.J. Cameron and R.A. Liebler, Lin. Alg. Appl. 46 (1982), 91–102) (and still open).