Hessian group

In mathematics, the Hessian group is a finite group of order 216, introduced by Jordan (1877) who named it for Otto Hesse. It may be represented as the group of affine transformations with determinant 1 of the affine plane over the field of 3 elements.[1] It also acts on the Hesse pencil of elliptic curves, and forms the automorphism group of the Hesse configuration of the 9 inflection points of these curves and the 12 lines through triples of these points.

The triple cover of this group is a complex reflection group, 3[3]3[3]3 or of order 648, and the product of this with a group of order 2 is another complex reflection group, 3[3]3[4]2 or of order 1296. It has a normal subgroup that is an elementary abelian group of order 32, and the quotient by this subgroup is isomorphic to the group SL2(3) of order 24.

References

  • Artebani, Michela; Dolgachev, Igor (2009), "The Hesse pencil of plane cubic curves", L'Enseignement Mathématique. Revue Internationale, 2e Série, 55 (3): 235–273, arXiv:math/0611590, doi:10.4171/lem/55-3-3, ISSN 0013-8584, MR 2583779
  • Coxeter, Harold Scott MacDonald (1956), "The collineation groups of the finite affine and projective planes with four lines through each point", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 20: 165–177, ISSN 0025-5858, MR 0081289
  • Grove, Charles Clayton (1906), The syzygetic pencil of cubics with a new geometrical development of its Hesse Group, Baltimore, Md.
  • Jordan, Camille (1877), "Mémoire sur les équations différentielles linéaires à intégrale algébrique.", Journal für die reine und angewandte Mathematik (in French), 84: 89–215, doi:10.1515/crll.1878.84.89, ISSN 0075-4102
  1. ↑ Hessian group on GroupNames
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