PSL(2,7) - Actions On Projective Spaces

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); the full group of possibly orientation-reversing projective linear automorphisms is instead the order 2 extension PGL(2,7), and the group of collineations of the projective line is the complete symmetric group of the points.

However, PSL(2,7) is also isomorphic to PSL(3,2) (= 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 = PSL(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 = PSL(3,2) 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.