# PSL(2,7)

The

projective special linear group *G* =

**PSL(2,7)** is a

finite group in

mathematics that has important applications in

algebra,

geometry, and

number theory. It is the

automorphism group of the

Klein quartic, and it is the second-smallest

nonabelian simple group, next to the

alternating group A

_{5} = PSL(2,5).

**Definition** Let SL(2,7) denote the group of all 2×2 matrices of determinant one over the finite field with 7 elements. Then *G* = PSL(2,7) is defined to be the quotient group SL(2,7) / {I,−I} obtained by identifying I and −I. In this article, we let *G* denote any group isomorphic to PSL(2,7).

*G* = PSL(2,7) has 168 elements. This can be seen by counting the possible columns; there are 7^{2} − 1 = 48 possibilities for the first column, then 7^{2} − 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 *S*_{3}, and in the latter case PSL(

*n*,

*q*) is isomorphic to

alternating group *A*_{4}. In fact, PSL(2,7) is the second smallest

nonabelian simple group, next to the

alternating group A

_{5} = PSL(2,5).

*G* = PSL(2,7) acts via linear fractional transformation on the

projective line **P**^{1}(7) over the field with 7 elements:

Every automorphism of **P**^{1}(7) arises in this way, and so *G* = PSL(2,7) can be thought of geometrically as the group of symmetries of the projective line **P**^{1}(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 Fano plane **P**^{2}(2) over the field with 2 elements:

Again, every automorphism of **P**^{2}(2) arises in this way, and so *G* = SL(3,2) can be thought of geometrically as the group of symmetries of the Fano plane.

The Klein quartic *x*^{3}*y* +

*y*^{3}*z* +

*z*^{3}*x* = 0 is a

Riemann surface, the unique curve of genus 3 over the

complex numbers **C**. The Klein quartic has automorphism group isomorphic to

*G*. The order 168 of

*G* is the maximum allowed for this genus. Klein's quartic pops up all over the place in mathematics, not least of which includes representation theory, homology theory, octonion multiplication,

Fermat's Last Theorem, and

Stark's theorem on imaginary quadratic number fields of class number 1!