Valentiner group

In mathematics, the Valentiner group is the perfect triple cover of the alternating group on 6 points, and is a group of order 1080. It was found by Herman Valentiner (1889) in the form of an action of A₆ on the complex projective plane, and was studied further by Wiman (1896).

All perfect alternating groups have perfect double covers. In most cases this is the universal central extension. The two exceptions are A₆ (whose perfect triple cover is the Valentiner group) and A₇, whose universal central extensions have centers of order 6.

Representations

• The alternating group A₆ acts on the complex projective plane, and Gerbaldi (1898) showed that the group acts on the 6 conics of Gerbaldi's theorem. This gives a homomorphism to PGL₃(C), and the lift of this to the triple cover GL₃(C) is the Valentiner group. This embedding can be defined over the field generated by the 15th roots of unity.
• The product of the Valentiner group with a group of order 2 is a 3-dimensional complex reflection group of order 2160 generated by 45 complex reflections of order 2. The invariants form a polynomial algebra with generators of degrees 6, 12, and 30.
• The Valentiner group has complex irreducible faithful group representations of dimension 3, 3, 3, 3, 6, 6, 9, 9, 15, 15.
• The Valentiner group can be represented as the monomial symmetries of the hexacode, the 3-dimensional subspace of F⁶
  ₄ spanned by (001111), (111100), and (0101ωω), where the elements of the finite field F₄ are 0, 1, ω, ω.
• The group PGL₃(F₄) acts on the 2-dimensional projective plane over F₄ and acts transitively on its hyperovals (sets of 6 points such that no three are on a line). The subgroup fixing a hyperoval is a copy of the alternating group A₆. The lift of this to the triple cover GL₃(F₄) of PGL₃(F₄) is the Valentiner group.
• Crespo & Hajto (2005) described the representations of the Valentiner group as a Galois group, and gave an order 3 differential equation with the Valentiner group as its differential Galois group.

References

• Coble, Arthur B. (1911), "The reduction of the sextic equation to the Valentiner form-problem", Mathematische Annalen, 70, Springer Berlin / Heidelberg: 337–350, doi:10.1007/BF01564501, ISSN 0025-5831
• Crass, Scott (1999), "Solving the sextic by iteration: a study in complex geometry and dynamics", Experimental Mathematics, 8 (3): 209–240, doi:10.1080/10586458.1999.10504401, ISSN 1058-6458, MR 1724156
• Crespo, Teresa; Hajto, Zbigniew (2005), "The Valentiner group as Galois group", Proceedings of the American Mathematical Society, 133 (1): 51–56, doi:10.1090/S0002-9939-04-07539-2, ISSN 0002-9939, MR 2085152
• Gerbaldi, Francesco (1898), "Sul gruppo semplice di 360 collineazioni piane", Mathematische Annalen, 50, Springer Berlin / Heidelberg: 473–476, doi:10.1007/BF01448080, ISSN 0025-5831
• Valentiner, H. (1889), "De endelige Transformations-gruppers Theori", Videnkabernes Selskabs Skrifter (in Danish), 6, Bianco Lunos
• Wiman, A. (1896), "Ueber eine einfache Gruppe von 360 ebenen Collineationen", Mathematische Annalen, 47, Springer Berlin / Heidelberg: 531–556, doi:10.1007/BF01445800, ISSN 0025-5831, JFM 27.0103.03