PSL(2,7)
Template:Short description In mathematics, the projective special linear group Template:Nowrap, isomorphic to Template:Nowrap, is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements, PSL(2, 7) is the smallest nonabelian simple group after the alternating group A5 with 60 elements, isomorphic to Template:Nowrap.
DefinitionEdit
The general linear group Template:Nowrap consists of all invertible 2×2 matrices over F7, the finite field with 7 elements. These have nonzero determinant. The subgroup Template:Nowrap consists of all such matrices with unit determinant. Then Template:Nowrap 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 this article, we let G denote any group that is isomorphic to Template:Nowrap.
PropertiesEdit
G = Template:Nowrap has 168 elements. This can be seen by counting the possible columns; there are Template:Nowrap possibilities for the first column, then Template:Nowrap possibilities for the second column. We must divide by Template:Nowrap to force the determinant equal to one, and then we must divide by 2 when we identify I and −I. The result is Template:Nowrap.
It is a general result that Template:Nowrap is simple for Template:Nowrap (q being some power of a prime number), unless Template:Nowrap or Template:Nowrap. Template:Nowrap is isomorphic to the symmetric group S3, and Template:Nowrap is isomorphic to alternating group A4. In fact, Template:Nowrap is the second smallest nonabelian simple group, after the alternating group Template:Nowrap.
The number of conjugacy classes and irreducible representations is 6. The sizes of conjugacy classes are 1, 21, 42, 56, 24, 24. The dimensions of irreducible representations 1, 3, 3, 6, 7, 8.
Character table
- <math>\begin{array}{r|cccccc}
& 1A_{1} & 2A_{21} & 4A_{42} & 3A_{56} & 7A_{24} & 7B_{24} \\ \hline
\chi_1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \chi_2 & 3 & -1 & 1 & 0 & \sigma & \bar \sigma \\ \chi_3 & 3 & -1 & 1 & 0 & \bar \sigma & \sigma \\ \chi_4 & 6 & 2 & 0 & 0 & -1 & -1 \\ \chi_5 & 7 & -1 &-1 & 1 & 0 & 0 \\ \chi_6 & 8 & 0 & 0 & -1 & 1 & 1 \\ \end{array},</math> where
- <math>\sigma = \frac{-1+i\sqrt{7}}{2}.</math>
The following table describes the conjugacy classes in terms of the order of an element in the class, the size of the class, the minimum polynomial of every representative in GL(3, 2), and the function notation for a representative in PSL(2, 7). Note that the classes 7A and 7B are exchanged by an automorphism, so the representatives from GL(3, 2) and PSL(2, 7) can be switched arbitrarily.
| Order | Size | Min Poly | Function |
|---|---|---|---|
| 1 | 1 | x + 1 | x |
| 2 | 21 | x2 + 1 | −1/x |
| 3 | 56 | x3 + 1 | 2x |
| 4 | 42 | x3 + x2 + x + 1 | 1/(3 − x) |
| 7 | 24 | x3 + x + 1 | x + 1 |
| 7 | 24 | x3 + x2 + 1 | x + 3 |
The order of group is Template:Nowrap, this implies existence of Sylow's subgroups of orders 3, 7 and 8. It is easy to describe the first two, they are cyclic, since any group of prime order is cyclic. Any element of conjugacy class 3A56 generates Sylow 3-subgroup. Any element from the conjugacy classes 7A24, 7B24 generates the Sylow 7-subgroup. The Sylow 2-subgroup is a dihedral group of order 8. It can be described as centralizer of any element from the conjugacy class 2A21. In the Template:Nowrap representation, a Sylow 2-subgroup consists of the upper triangular matrices.
This group and its Sylow 2-subgroup provide a counter-example for various normal p-complement theorems for Template:Nowrap.
Actions on projective spacesEdit
G = Template:Nowrap acts via linear fractional transformation on the projective line P1(7) over the field with 7 elements:
- <math>\text{For } \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{PSL}(2, 7) \text{ and } x \in \mathbf{P}^1\!(7),\ \gamma \cdot x = \frac{ax+b}{cx+d} .</math>
Every orientation-preserving automorphism of P1(7) arises in this way, and so Template:Nowrap can be thought of geometrically as a group of symmetries of the projective line P1(7); the full group of possibly orientation-reversing projective linear automorphisms is instead the order 2 extension Template:Nowrap, and the group of collineations of the projective line is the complete symmetric group of the points.
However, Template:Nowrap is also isomorphic to Template:Nowrap (Template:Nowrap), the special (general) linear group of 3×3 matrices over the field with 2 elements. In a similar fashion, Template:Nowrap acts on the projective plane P2(2) over the field with 2 elements — also known as the Fano plane:
- For <math> \gamma = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \in \text{PSL}(3, 2)\ \ </math> and <math>\
\ \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \in \mathbf{P}^2\!(2),\ \ \gamma \ \cdot \ \mathbf{x} = \begin{pmatrix} ax+by+cz \\ dx+ey+fz \\ gx+hy+iz \end{pmatrix}</math>
Again, every automorphism of P2(2) arises in this way, and so Template:Nowrap can be thought of geometrically as the symmetry group of this projective plane. The Fano plane can be used to describe multiplication of octonions, so G acts on the set of octonion multiplication tables.
Symmetries of the Klein quarticEdit
The Klein quartic is the projective variety over the complex numbers C defined by the quartic polynomial
- x3y + y3z + z3x = 0.
It is a compact Riemann surface of genus Template:Nowrap, and is the only such surface for which the size of the conformal automorphism group attains the maximum of Template:Nowrap. This bound is due to the Hurwitz automorphisms theorem, which holds for all Template:Nowrap. Such "Hurwitz surfaces" are rare; the next genus for which any exist is Template:Nowrap, and the next after that is Template:Nowrap.
As with all Hurwitz surfaces, the Klein quartic can be given a metric of constant negative curvature and then tiled with regular (hyperbolic) heptagons, as a quotient of the order-3 heptagonal tiling, with the symmetries of the surface as a Riemannian surface or algebraic curve exactly the same as the symmetries of the tiling. For the Klein quartic this yields a tiling by 24 heptagons, and the order of G is thus related to the fact that Template:Nowrap. Dually, it can be tiled with 56 equilateral triangles, with 24 vertices, each of degree 7, as a quotient of the order-7 triangular tiling.
Klein's quartic arises in many fields of 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.
Mathieu groupEdit
Template:Nowrap is a maximal subgroup of the Mathieu group M21; the groups M21 and M24 can be constructed as extensions of Template:Nowrap. These extensions can be interpreted in terms of the tiling of the Klein quartic, but are not realized by geometric symmetries of the tiling.<ref name="richter">Template:Harv</ref>
Permutation actionsEdit
The group Template:Nowrap acts on various finite sets:
- In its original interpretation as Template:Nowrap, orientation-preserving linear automorphisms of the projective line P1(F7), it acts transitively on the 8 points with a stabilizer of order 21 fixing a given point. It also acts 2-transitively with stabilizer of order 3 on each pair of points; and it has two orbits on triples of points, with trivial stabilizer on each triple. (The larger group Template:Nowrap acts sharply 3-transitively.)
- Interpreted as Template:Nowrap, linear automorphisms of the Fano plane P2(F2), it acts 2-transitively on the 7 points, with stabilizer of order 24 fixing each point, and stabilizer of order 4 fixing each pair of points.
- Interpreted as automorphisms of a tiling of the Klein quartic, it acts transitively on the 24 vertices (or dually, 24 heptagons), with stabilizer of order 7 (corresponding to a rotation about the vertex/heptagon).
- Interpreted as a subgroup of the Mathieu group M21, the subgroup acts non-transitively on 21 points.
ReferencesEdit
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