Symmetric designs with bruck subdesigns

Abstract

IfP is a finite projective plane of ordern with a proper subplaneQ of orderm which is not a Baer subplane, then a theorem of Bruck [Trans. AMS 78(1955), 464–481] asserts thatnm 2+m. If the equalityn=m 2+m were to occur thenP would be of composite order andQ should be called a Bruck subplane. It can be shown that if a projective planeP contains a Bruck subplaneQ, then in factP contains a designQ′ which has the parameters of the lines in a three dimensional projective geometry of orderm. A well known scheme of Bruck suggests using such aQ′ to constructP. Bruck’s theorem readily extends to symmetric designs [Kantor, Trans. AMS 146 (1969), 1–28], hence the concept of a Bruck subdesign. This paper develops the analoque ofQ′ and shows (by example) that the analogous construction scheme can be used to find symmetric designs.

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Baker, R.D. Symmetric designs with bruck subdesigns. Combinatorica 2, 103–109 (1982). https://doi.org/10.1007/BF02579308

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AMS (1980) subject classification

  • 05 B 30
  • 05 B 25
  • 50 D 35
  • 62 K 10