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Hessian Group
In mathematics, the Hessian group is a finite group of order 216, introduced by 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.Hessian group oGroupNames/ref> 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. 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 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group
Finite is the opposite of infinite. It may refer to: * Finite number (other) * 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 for person and/or tense or aspect * "Finite", a song by Sara Groves from the album '' Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb is a verb that is not capable of serving as the main verb in an independent clause * a non-finite clause In linguistics, a non-finite clause is a dependent or embedded clause that represen ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Otto Hesse
Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician. Hesse was born in Königsberg, Kingdom of Prussia, Prussia, and died in Munich, Kingdom of Bavaria, Bavaria. He worked mainly on algebraic invariants, and geometry. The Hessian matrix, the Hesse normal form, the Hesse configuration, the Hessian group, Hessian pairs, Hesse's theorem, Hesse pencil, and the Hesse transfer principle are named after him. Many of Hesse's research findings were presented for the first time in ''Crelle's Journal'' or Hesse's textbooks.MacTutor History of Mathematics archive and Complete Dictionary of Scientific Biography Life Hesse was born in Königsberg (today Kaliningrad) as the son of Johann Gottlieb Hesse, a businessman and brewery owner and his wife Anna Karoline Reiter (1788–1865). He studied in his hometown at the University of Königsberg, Albertina under Carl Gustav Jacob Jacobi. Among his teachers were count Friedrich Wilhelm Bessel and Friedrich Julius Ric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. If is the point set of an affine space, then every affine transformation on can be repre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 groups are a particular kind of ''p''-group. The case where ''p'' = 2, i.e., an elementary abelian 2-group, is sometimes called a Boolean group. Every elementary abelian ''p''-group is a vector space over the prime field with ''p'' elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form (Z/''p''Z)''n'' for ''n'' a non-negative integer (sometimes called the group's ''rank''). Here, Z/''p''Z denotes the cyclic group of order ''p'' (or equivalently the integers mod ''p''), and the superscript notation means the ''n''-fold direct product of groups. In ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 in the family is determined by a pair of parameter values (\lambda,\mu) (not both zero) and consists of the points in the plane whose homogeneous coordinates (x,y,z) satisfy the equation for those parameters. Multiplying both \lambda and \mu by the same scalar does not change the curve, so there is only one degree of freedom in selecting a curve from the pencil, but the two-parameter form given above allows either \lambda or \mu (but not both) to be set to zero. Each curve in the pencil passes through the nine points of the complex projective plane whose homogeneous coordinates are some permutation of 0, –1, and a cube root of unity. There are three roots of unity, and six permutations per root, giving 18 choices for the homogeneous coor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 group of invertible linear transformations from ''X'' to itself (the general linear group of ''X''). If instead ''X'' is a group, then its automorphism group \operatorname(X) is the group consisting of all group automorphisms of ''X''. Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups are studied in a general way in the field of category theory. Examples If ''X'' is a set with no additional structure, then any bijection from ''X'' to itself is an automorphism, and hence the automorphism group of ''X'' in this case is precisely the symmetric group of ''X''. If the set ''X'' has additional struct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the set of inflection points of an elliptic curve, but it has no realization in the Euclidean plane. Description The Hesse configuration has the same incidence relations as the lines and points of the affine plane over the field of 3 elements. That is, the points of the Hesse configuration may be identified with ordered pairs of numbers modulo 3, and the lines of the configuration may correspondingly be identified with the triples of points satisfying a linear equation . Alternatively, the points of the configuration may be identified by the squares of a tic-tac-toe board, and the lines may be identified with the lines and broken diagonals of the board. Each point belongs to four lines: in the tic tac toe interpretation of the configura ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 in the study of the invariant theory of polynomial rings. In the mid-20th century, they were completely classified in work of Shephard and Todd. Special cases include the symmetric group of permutations, the dihedral groups, and more generally all finite real reflection groups (the Coxeter groups or Weyl groups, including the symmetry groups of regular polyhedra). Definition A (complex) reflection ''r'' (sometimes also called ''pseudo reflection'' or ''unitary reflection'') of a finite-dimensional complex vector space ''V'' is an element r \in GL(V) of finite order that fixes a complex hyperplane pointwise, that is, the ''fixed-space'' \operatorname(r) := \operatorname(r-\operatorname_V) has codimension 1. A (finite) complex reflectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Für Die Reine Und Angewandte Mathematik
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by August Leopold Crelle (Berlin) in 1826 and edited by him until his death in 1855. It was one of the first major mathematical journals that was not a proceedings of an academy. It has published many notable papers, including works of Niels Henrik Abel, Georg Cantor, Gotthold Eisenstein, Carl Friedrich Gauss and Otto Hesse. It was edited by Carl Wilhelm Borchardt from 1856 to 1880, during which time it was known as ''Borchardt's Journal''. The current editor-in-chief is Rainer Weissauer (Ruprecht-Karls-Universität Heidelberg) Past editors * 1826–1856 August Leopold Crelle * 1856–1880 Carl Wilhelm Borchardt * 1881–1888 Leopold Kronecker, Karl Weierstrass * 1889–1892 Leopold Kronecker * 1892–1902 Lazarus Fuchs * 1903–1928 Kurt Hens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
