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Three-Dimensional Classical Groups Acting on Polytopes

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Let V be a three-dimensional vector space over a finite field. We show that any irreducible subgroup of GL(V) that arises as the automorphism group of an abstract regular polytope preserves a nondegenerate symmetric bilinear form on V. In particular, the only classical groups on V that arise as automorphisms of such polytopes are the orthogonal groups.


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Correspondence to Peter A. Brooksbank.

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Brooksbank, P.A., Vicinsky, D.A. Three-Dimensional Classical Groups Acting on Polytopes. Discrete Comput Geom 44, 654–659 (2010). https://doi.org/10.1007/s00454-009-9212-0

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  • Abstract regular polytope
  • String C-group
  • Classical group