Abstract
For any odd integern ≥3 and prime powerq, it is known thatPG(n−1, q2) can be partitioned into pairwise disjoint subgeometries isomorphic toPG(n−1, q) by taking point orbits under an appropriate subgroup of a Singer cycle ofPG(n−1, q2). In this paper, we construct Baer subgeometry partitions of these spaces which do not arise in the classical manner. We further illustrate some of the connections between Baer subgeometry partitions and several other areas of combinatorial interest, most notably projective sets and flagtransitive translation planes.
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References
J. André, Uber nicht-Desarguesche Ebenen mit transitiver Translationsgruppe,Math. Z. 60 (1954), 156–186.
R.D. Baker andG.L. Ebert, Regulus-free spreads of PG(3, q),Des. Codes Cryptogr. 8 (1996), 79–89.
R.D. Baker andG.L. Ebert, Two-dimensional flag-transitive planes revisited,Geom. Dedicata 63 (1996), 1–15.
R.C. Bose, On a representation of the Baer subplanes of the Desarguesian plane PG(2, q2) in a projective five dimensional space PG(5,q), inTeorie Combinatorie, Volume I, pages 381–391, Accademia Nazionale dei Lincei, Rome. 1973.
R.H. Bruck, Quadratic extensions of cyclic planes.Proc. Sympos. Appl. Math. 10 (1960), 15–44.
A.R. Calderbank andW.M. Kantor, The geometry of two-weight codes,Bull. London Math. Soc. 18 (1986), 97–122.
J. Cannon andC. Playoust,An Introduction to MAGMA, University of Sydney Press, Sydney, 1993.
G.L. Ebert, Partitioning problems and flag-transitive planes,Rend. Circ. Mat. Palermo (II)53 (1998). 27–44.
H. Gevaert, N.L. Johnson andJ.A. Thas, Spreads covered by reguli,Simon Stevin 62 (1988), 51–62.
J.W.P. Hirschfeld,Projective Geometries over Finite Fields, Second edition, Oxford University Press, Oxford, 1998.
J.W.P. Hirschfeld andJ.A. Thas,General Galois Geometries. Oxford University Press, Oxford, 1991.
W.M. Kantor, 2-Transitive and flag-transitive designs, InCoding Theory, Design Theory, Group Theory, Wiley-Interscience Publications, pages 13–30, Wiley, New York, 1993.
W.M. Kantor, Two families of flag-transitive affine planes,Geom. Dedicata 41 (1992), 191–200.
W.M. Kantor andM.E. Williams, New flag-transitive affine planes of even order,J. Combin. Theory Ser. A 74 (1996), 1–13.
R. Mathon andN. Hamilton, Baer partitions of small order projective planes,J. Combin. Math. Combin. Comput. 29 (1999), 87–94.
M.L. Narayana Rao andK. Kuppuswamy Rao, A new flag transitive affine plane of order 27,Proc. Amer. Math. Soc. 59 (1976), 337–345.
T. Penttila andG.F. Royle, Sets of type (m, n) in the affine and projective planes of order nine,Des. Codes Cryptogr. 6 (1995), 229–245.
C. Suetake, Flag transitive planes of order qn with a long cycle ℓ∞ as a collineation,Graphs Combin. 7 (1991), 183–195.
J. Überberg, Projective planes and dihedral groups,Discrete Math. 174 (1997), 337–345.
P. Yff, On subplane partitions of a finite projective plane,J. Combin. Theory Ser. A 22 (1977), 118–122.
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Baker, R.D., Dover, J.M., Ebert, G.L. et al. Baer subgeometry partitions. J Geom 67, 23–34 (2000). https://doi.org/10.1007/BF01220294
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DOI: https://doi.org/10.1007/BF01220294