Local Recognition Of Non-Incident Point-Hyperplane Graphs

Let ℙ be a projective space. By H(ℙ) we denote the graph whose vertices are the non-incident point-hyperplane pairs of ℙ, two vertices (p,H) and (q,I) being adjacent if and only if pI and qH. In this paper we give a characterization of the graph H(ℙ) (as well as of some related graphs) by its local structure. We apply this result by two characterizations of groups G with PSL n (\(\Bbb F\))≤G≤PGL n (\(\Bbb F\)), by properties of centralizers of some (generalized) reflections. Here \(\Bbb F\) is the (skew) field of coordinates of ℙ.

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Correspondence to Arjeh M. Cohen.

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Cohen, A.M., Cuypers, H. & Gramlich, R. Local Recognition Of Non-Incident Point-Hyperplane Graphs. Combinatorica 25, 271–296 (2005). https://doi.org/10.1007/s00493-005-0016-4

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Mathematics Subject Classification (2000):

  • 05C25
  • 20D06
  • 20E42
  • 51E25