ABSTRACT
Conversely, the specification of a transitive effective action of the additive group of L on the set A determines on A the structure of an affine space with associated space L.
Proof. It follows from axioms a) and b) that for any 1 E L and a E A the equation ti(x) = a has the solution x = a + (—I), so that all ti are surjective. If t1(a) = tt(b), then having found by axiom c) a vector m E L such that b = a + m, we obtain
a + / = (a + m) + / = (a + I) + m. But a + / = (a + /) + 0, and therefore from the uniqueness condition in axiom c) it follows that in = 0, so that a = b. Therefore all the t, are injective.
