Definition:Affine Space/Group Action

Definition
Let $K$ be a field.

Let $\struct {V, +_V, \circ}$ be a vector space over $K$.

Let $\EE$ be a set.

Let $\phi: \EE \times V \to \EE$ be a free and transitive group action of $\struct {V, +_V}$ on $\EE$.

Then the ordered pair $\struct {\EE, \phi}$ is an affine space.

Also see

 * Equivalence of Definitions of Affine Space