PSL(2,7)

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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. 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 second-smallest nonabelian simple group after the alternating group A₅ on five letters with 60 elements, or the isomorphic PSL(2,5).

1 Definition
2 Properties
3 Actions on projective spaces
4 Symmetries of the Klein quartic
5 External links

[edit] Definition

The general linear group GL(2,7) consists of all 2×2 matrices over F₇, the finite field with 7 elements, which have nonzero determinant. The subgroup SL(2,7) consists of all such matrices with unit determinant. 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 this article, we let G denote any group isomorphic to PSL(2,7).

[edit] 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, unless (n, q) = (2,2) or (2,3). 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 fact, PSL(2,7) is the second smallest nonabelian simple group, next to the alternating group A₅ = PSL(2,5).

[edit] 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:

$\mbox{For } \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mbox{PSL}(2,7) \mbox{ and } x \in \mathbb{P}^1(7),\ \gamma \cdot x = \frac{ax+b}{cx+d}$

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).

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 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:

$\mbox{For } \gamma = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \in \mbox{SL}(3,2) \mbox{ and } \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \in \mathbb{P}^2(2),\ \gamma \ \cdot \ \mathbf{x} = \begin{pmatrix} ax+by+cz \\ dx+ey+fz \\ gx+hy+iz \end{pmatrix}$

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 the eightfold cube.) The Fano plane can be used to describe multiplication of octonions, so G acts on the set of octonion multiplication tables.

[edit] 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. Its group of conformal transformations is none other than G. No other curve of genus 3 has as many conformal transformations. In fact, Hurwitz proved that a curve of genus g has at most

84(g − 1) conformal transformations

(for g > 1).

The Klein quartic can be given a metric of constant negative curvature and then tiled with 24 regular heptagons. 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.