In mathematics, the projective special linear group PSL(2,7) is a finite simple group that has important applications in algebra, geometry, and number theory. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, especially in area of Algebra called Group theory, the projective linear group (also known as the projective general linear group In Mathematics, a finite group is a group which has finitely many elements SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each With 168 elements PSL(2,7) is the second-smallest nonabelian simple group after the alternating group A5 on five letters with 60 elements, or the isomorphic PSL(2,5). In Mathematics, a nonabelian group, also sometimes called a non-commutative group, is a group ( G, *) such that there are at least two elements In Mathematics, an alternating group is the group of Even permutations of a Finite set.
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Definition
The general linear group GL(2,7) consists of all invertible 2×2 matrices over F7, the finite field with 7 elements. In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements These have nonzero determinant. The subgroup SL(2,7) consists of all such matrices with unit determinant. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n Then PSL(2,7) is defined to be the quotient group
- SL(2,7) / {I,−I}
obtained by identifying I and −I, where I is the identity matrix. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main In this article, we let G denote any group isomorphic to PSL(2,7).
Properties
G = PSL(2,7) has 168 elements. This can be seen by counting the possible columns; there are 72 − 1 = 48 possibilities for the first column, then 72 − 7 = 42 possibilities for the second column. We must divide by 7 − 1 = 6 to force the determinant equal to one, and then we must divide by 2 when we identify I and −I. The result is (48*42) / (6*2) = 168.
It is a general result that PSL(n, q) is simple for n ≥ 2, q ≥ 2 (q being some power of a prime number), unless (n, q) = (2,2) or (2,3). SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning In the former case, PSL(n, q) is isomorphic to the symmetric group S3, and in the latter case PSL(n, q) is isomorphic to alternating group A4. In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In Mathematics, an alternating group is the group of Even permutations of a Finite set. In fact, PSL(2,7) is the second smallest nonabelian simple group, next to the alternating group A5 = PSL(2,5). In Mathematics, a nonabelian group, also sometimes called a non-commutative group, is a group ( G, *) such that there are at least two elements In Mathematics, an alternating group is the group of Even permutations of a Finite set.
Actions on projective spaces
G = PSL(2,7) acts via linear fractional transformation on the projective line P1(7) over the field with 7 elements:
Every orientation-preserving automorphism of P1(7) arises in this way, and so G = PSL(2,7) can be thought of geometrically as a group of symmetries of the projective line P1(7). Möbius transformations should not be confused with the Möbius transform or the Möbius function. In Mathematics, a projective line is a one-dimensional Projective space.
However, PSL(2,7) is also isomorphic to SL(3,2) (= GL(3,2)), the special (general) linear group of 3×3 matrices over the field with 2 elements. In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in In a similar fashion, G = SL(3,2) acts on the projective plane P2(2) over the field with 2 elements — also known as the Fano plane:
Again, every automorphism of P2(2) arises in this way, and so G = SL(3,2) can be thought of geometrically as the symmetry group of this projective plane. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is The Fano plane can be used to describe multiplication of octonions, so G acts on the set of octonion multiplication tables. In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real
Symmetries of the Klein quartic
The Klein quartic
- x3y + y3z + z3x = 0
is a Riemann surface, the most symmetrical curve of genus 3 over the complex numbers C. In Hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Its group of conformal transformations is none other than G. No other curve of genus 3 has as many conformal transformations. In fact, Adolf Hurwitz proved that a curve of genus g has at most
- 84(g − 1) conformal transformations
(for g > 1). Adolf Hurwitz ( 26 March 1859 - 18 November 1919) (ˈadɒlf ˈhurvits was a German mathematician and was described by Jean-Pierre
The Klein quartic can be given a metric of constant negative curvature and then tiled with 24 regular heptagons. In Mathematics, the Poincaré metric, named after Henri Poincaré, is the Metric tensor describing a two-dimensional surface of constant negative Curvature General properties These properties apply to both convex and star regular polygons Construction A regular heptagon is not constructible with Compass and straightedge but is constructible with a marked Ruler and compass The order of G is thus related to the fact that 24 x 7 = 168.
Klein's quartic pops up all over the place in mathematics, including representation theory, homology theory, octonion multiplication, Fermat's last theorem, and Stark's theorem on imaginary quadratic number fields of class number 1. Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like In Number theory, a branch of Mathematics, the Stark–Heegner theorem states precisely which quadratic imaginary number fields admit unique factorisation
External links
- The Eightfold Way: the Beauty of Klein's Quartic Curve (Silvio Levy, ed.)
- This Week's Finds in Mathematical Physics - Week 214 (John Baez)
- The Klein Quartic in Number Theory (Noam Elkies)
- Octonion Products and Lattices (Tony Smith)
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