Tactical Decompositions of Steiner Systems and Orbits of Projective Groups
Keldon Drudge
DOI: 10.1023/A:1026535809272
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, 
) 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 = 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).
) 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 = 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).Pages: 123–130
Keywords: tactical decompositions; partial spread; tightset; Steiner systems; projective group
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References
1. P.J. Cameron, Permutation Groups, Cambridge University Press, Cambridge, 1999.
2. P.J. Cameron and R.A. Liebler, “Tactical decompositions and orbits of projective groups,” Lin. Alg. Appl. 46 (1982), 91-102.
3. P. Dembowski, Finite Geometries, Springer-Verlag, Berlin-Heidelberg, 1968.
4. K. Drudge, “Extremal sets in projective and polar spaces,” Ph.D. Thesis, University of Western Ontario, 1998.
5. D.G. Glynn, “On a set of lines of PG(3, q) corresponding to a maximal cap contained in the Klein quadric of PG(5, q),” Geom. Ded. 26 (1988), 273-280.
6. W.H. Haemers, Eigenvalue Techniques in Design Theory and Graph Theory, Mathematisch Centrum, Amsterdam,
1980. Mathematical Centre Tracts #121.
7. H. L\ddot uneburg, Translation Planes, Springer-Verlag, Berlin-Heidelberg, 1980.
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2. P.J. Cameron and R.A. Liebler, “Tactical decompositions and orbits of projective groups,” Lin. Alg. Appl. 46 (1982), 91-102.
3. P. Dembowski, Finite Geometries, Springer-Verlag, Berlin-Heidelberg, 1968.
4. K. Drudge, “Extremal sets in projective and polar spaces,” Ph.D. Thesis, University of Western Ontario, 1998.
5. D.G. Glynn, “On a set of lines of PG(3, q) corresponding to a maximal cap contained in the Klein quadric of PG(5, q),” Geom. Ded. 26 (1988), 273-280.
6. W.H. Haemers, Eigenvalue Techniques in Design Theory and Graph Theory, Mathematisch Centrum, Amsterdam,
1980. Mathematical Centre Tracts #121.
7. H. L\ddot uneburg, Translation Planes, Springer-Verlag, Berlin-Heidelberg, 1980.
8. S.E. Payne, “Tight pointsets in finite generalized quadrangles I,” Congressus Numerantium 60 (1987), 243-260; II, Congressus Numerantium 77 (1990), 31-41.
9. T. Penttila, “Collineations and configurations in projective spaces,” Ph.D. Thesis, Oxford University, 1985.