Combinatorial description of the affine group

The affine group A was defined with reference to the metric in the Euclidean plane E, and was described in terms of real matrices. However, the affine group does not preserve the Euclidean metric, and it is possible to give a characterization of A that makes no reference to this metric. We introduce the temporary notation L for the group of all bisections from E to E that map lines to lines. We shall show that, in fact, A = L.

We outline the proof before turning to details. It is clear that A⊆L, whence it remains only to prove that L ⊆ A. Now this inclusion reduces easily to the assertion that if a in L fixes two distinct points of a line l, then α fixes all points of the line. Introducing coordinates, we may suppose that the line l is the x-axis and that α fixes (0,0) and (1,0). It is clear that a maps l bijectively to itself, whence the equation (x,0)α = (xγ,0) defines a bijection γ of R The next step is to show that γ is an automorphism of R, γ ε Aut R, in the sense that, for all x, y ε R, (x + y)γ = xγ + yγ and (xy)γ = (xγ)(yγ). The final step is to show that the identity is the only automorphism of R, that is, if γ ε Aut R, then xγ = x for all x ε R.