In mathematics, the Hessian group is a
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
of order 216, introduced by who named it for
Otto Hesse
Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician. Hesse was born in Königsberg, Prussia, and died in Munich, Bavaria. He worked mainly on algebraic invariants, and geometry. The Hessian matrix, the Hesse nor ...
. It may be represented as the group of
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
s with determinant 1 of the affine plane over the field of 3 elements.
[Hessian group o]
GroupNames
/ref> It has a normal subgroup that is an elementary abelian group
In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian grou ...
of order 32, and the quotient by this subgroup is isomorphic to the group SL2(3) of order 24. It also acts on the Hesse pencil In mathematics, the syzygetic pencil or Hesse pencil, named for Otto Hesse, is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the equation
:\lambda(x^3+y^3+z^3) + \mu xyz =0.
Each curve ...
of elliptic curves, and forms the automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the Hesse configuration
In geometry, the Hesse configuration, introduced by Colin Maclaurin and studied by , is a configuration of 9 points and 12 lines with three points per line and four lines through each point. It can be realized in the complex projective plane as ...
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 In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise.
Complex reflection groups arise ...
, 3 sub>3 sub>3 or of order 648, and the product of this with a group of order 2 is another complex reflection group, 3 sub>3 sub>2 or of order 1296.
References
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External links
{{math-stub
Finite groups